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  <ul class="toc">
  <li class="tocentry"><span class="ref toc here">Normal ordering for deformed boson operators and
operator-valued deformed Stirling numbers</span>
  <ul class="toc">
  <li class="tocentry"><a href="#S1" title="§1. Introduction in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref toc">§1 Introduction</a></li>
  <li class="tocentry"><a href="#S2" title="§2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref toc">§2 Stirling and deformed Stirling numbers</a></li>
  <li class="tocentry"><a href="#S3" title="§3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref toc">§3 Some algebraic properties of deformed boson operators</a></li>
  <li class="tocentry"><a href="#S4" title="§4. Normal ordering of powers of the deformed number operator in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref toc">§4 Normal ordering of powers of the deformed number operator</a></li>
  <li class="tocentry"><a href="#S5" title="§5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref toc">§5 Operator-valued deformed Stirling numbers</a></li>
  <li class="tocentry"><a href="#S6" title="§6. A generating function for the deformed Stirling numbers&#10;of the first kind in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref toc">§6 A generating function for the deformed Stirling numbers
of the first kind</a></li>
  <li class="tocentry"><a href="#S7" title="§7. Discussion in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref toc">§7 Discussion</a></li>
  <li class="tocentry"><a href="#bib" title="References in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref toc">References</a></li>
    </ul></li>
    </ul>
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    <h1 class="title document-title">Normal ordering for deformed boson operators and
operator-valued deformed Stirling numbers</h1>
    <div class="creator author">
    <div class="personname"> Jacob Katriel
<m:math display="inline"><m:semantics><m:none/><m:annotation-xml encoding="MathML-Content"><m:ci/></m:annotation-xml></m:semantics></m:math> and Maurice Kibler <br class="break"/><span style="" class="text slanted small">Institut de Physique Nucléaire de Lyon </span> <br class="break"/><span style="" class="text slanted small">IN2P3-CNRS et Université Claude Bernard</span> <br class="break"/><span style="" class="text slanted small">43 Boulevard du 11 Novembre 1918</span> <br class="break"/><span style="" class="text slanted small">F-69622 Villeurbanne Cedex, France</span> </div>
  Permanent address:
Department of Chemistry, Technion - Israel Institute of Technology,
Haifa 32000, Israel. email: chr09kt@technion.email: Kibler@ipnl.in2p3.fr.</div>
  
    <div class="abstract">
      <h6>Abstract. </h6>
      
    <p class="p">The normal ordering formulae for powers of the boson number operator
<m:math display="inline"><m:semantics><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>
are extended to deformed bosons. It is found that for the “M-type”
deformed bosons, which satisfy <m:math display="inline"><m:semantics><m:mrow><m:mrow><m:mrow><m:mi>a</m:mi><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mrow><m:mo>-</m:mo><m:mrow><m:mi>q</m:mi><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup><m:mo>⁢</m:mo><m:mi>a</m:mi></m:mrow></m:mrow><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:minus/><m:apply><m:times/><m:ci>a</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply></m:apply><m:apply><m:times/><m:ci>q</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>a</m:ci></m:apply></m:apply><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math>,
the extension involves a set of deformed Stirling numbers which
replace the Stirling numbers occurring in the conventional case.
On the other hand, the deformed Stirling numbers which
have to be introduced in the case of the “P-type” deformed bosons,
which satisfy <m:math display="inline"><m:semantics><m:mrow><m:mrow><m:mrow><m:mi>a</m:mi><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mrow><m:mo>-</m:mo><m:mrow><m:mi>q</m:mi><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup><m:mo>⁢</m:mo><m:mi>a</m:mi></m:mrow></m:mrow><m:mo>=</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:msup></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:minus/><m:apply><m:times/><m:ci>a</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply></m:apply><m:apply><m:times/><m:ci>q</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>a</m:ci></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>,
are found to depend on the operator <m:math display="inline"><m:semantics><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>. This distinction
between the two types of deformed bosons is in harmony with earlier
observations made in the context of a study of the extended
Campbell-Baker-Hausdorff formula.</p>
  
    <p class="p">Published in J. Phys. A : Math. Gen. 25 (1992) 2683-2691.</p>
  
    </div>
  
    <div class="section" id="S1">
    <h2 class="title section-title"><span class="refnum"><span class="reftype">§ </span>1. </span>Introduction</h2>
  <div class="para" id="S1.p1">
    <p class="p">The transformation of a second-quantized operator
into a normally ordered form, in which each
term is written with the creation operators preceding the annihilation
operators, has been found to simplify quantum mechanical calculations in
a large and varied range of situations. Techniques for the
accomplishment of this ordering  have been developed and are widely
utilized [1,2]. A particular
subclass of problems and techniques involves situations in which the
operators of interest commute with the number operator.
More specifically,
one is interested in transforming an operator  which is  a function  of
the number operator into a normally ordered form,
or transforming an operator
each of whose terms has an equal number of creation and annihilation
operators corresponding to each degree of freedom, into an equivalent
operator expressed in terms of the number operator only.</p>
  </div>

  <div class="para" id="S1.p2">
    <p class="p">In the present article we consider the corresponding problem for the
deformed bosons which have been investigated very extensively in the
last three years [3,4]
in connection with the recent interest in
the properties and
applications of quantum groups.</p>
  </div>

    </div>
  
    <div class="section" id="S2">
    <h2 class="title section-title"><span class="refnum"><span class="reftype">§ </span>2. </span>Stirling and deformed Stirling numbers</h2>
  <div class="para" id="S2.p1">
    <p class="p">The Stirling numbers of the first <m:math display="inline"><m:semantics><m:mfenced open="(" close=")"><m:mi>s</m:mi></m:mfenced><m:annotation-xml encoding="MathML-Content"><m:ci>s</m:ci></m:annotation-xml></m:semantics></m:math>
and second <m:math display="inline"><m:semantics><m:mfenced open="(" close=")"><m:mi>S</m:mi></m:mfenced><m:annotation-xml encoding="MathML-Content"><m:ci>S</m:ci></m:annotation-xml></m:semantics></m:math> kinds were introduced in
connection with the expression for a descending product of a
variable <m:math display="inline"><m:semantics><m:mi>x</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>x</m:ci></m:annotation-xml></m:semantics></m:math> as a linear combination of integral and positive powers of
that variable, and the inverse relation, respectively <cite class="cite">[<a href="#bib.bib5" title="" class="ref">5</a>]</cite></p>
  <table class="equation" id="S2.E1">
      
      
	<tr valign="baseline" class="equation" id="S2.E1">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mi>x</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>x</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mi mathvariant="normal">⋯</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>x</m:mi><m:mo>-</m:mo><m:mi>k</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>m</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>k</m:mi></m:mover><m:mi>s</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msup><m:mi>x</m:mi><m:mi>m</m:mi></m:msup></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:ci>x</m:ci><m:apply><m:minus/><m:ci>x</m:ci><m:cn>1</m:cn></m:apply><m:ci>⋯</m:ci><m:apply><m:plus/><m:apply><m:minus/><m:ci>x</m:ci><m:ci>k</m:ci></m:apply><m:cn>1</m:cn></m:apply></m:apply><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:times/><m:ci>s</m:ci><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>x</m:ci><m:ci>m</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">1</span>)</td>
	</tr>
      
      </table><table class="equation" id="S2.E2">
      
      
	<tr valign="baseline" class="equation" id="S2.E2">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msup><m:mi>x</m:mi><m:mi>m</m:mi></m:msup><m:mo>=</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>k</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>m</m:mi></m:mover><m:mi>S</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mi>x</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>x</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mi mathvariant="normal">⋯</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>x</m:mi><m:mo>-</m:mo><m:mi>k</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>x</m:ci><m:ci>m</m:ci></m:apply><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:times/><m:ci>S</m:ci><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply><m:ci>x</m:ci><m:apply><m:minus/><m:ci>x</m:ci><m:cn>1</m:cn></m:apply><m:ci>⋯</m:ci><m:apply><m:plus/><m:apply><m:minus/><m:ci>x</m:ci><m:ci>k</m:ci></m:apply><m:cn>1</m:cn></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">2</span>)</td>
	</tr>
      
      </table>
    <p class="p">Using these defining relations it is easy to show that the Stirling
numbers satisfy the recurrence relations</p>
  <table class="equation" id="S2.E3">
      
      
	<tr valign="baseline" class="equation" id="S2.E3">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mi>s</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:mi>k</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mrow><m:mi>s</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mrow><m:mi>m</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mrow></m:mfenced></m:mrow><m:mo>-</m:mo><m:mrow><m:mi>k</m:mi><m:mo>⁢</m:mo><m:mi>s</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced></m:mrow></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:ci>s</m:ci><m:apply><m:interval closure="open"/><m:apply><m:plus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply><m:ci>m</m:ci></m:apply></m:apply><m:apply><m:minus/><m:apply><m:times/><m:ci>s</m:ci><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:apply><m:minus/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply></m:apply></m:apply><m:apply><m:times/><m:ci>k</m:ci><m:ci>s</m:ci><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">3</span>)</td>
	</tr>
      
      </table>
    <p class="p">and</p>
  <table class="equation" id="S2.E4">
      
      
	<tr valign="baseline" class="equation" id="S2.E4">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:mi>S</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:mi>m</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mrow><m:mi>S</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mrow></m:mfenced></m:mrow><m:mo>+</m:mo><m:mrow><m:mi>k</m:mi><m:mo>⁢</m:mo><m:mi>S</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow></m:mrow></m:mrow><m:mo>,</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:ci>S</m:ci><m:apply><m:interval closure="open"/><m:apply><m:plus/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply><m:ci>k</m:ci></m:apply></m:apply><m:apply><m:plus/><m:apply><m:times/><m:ci>S</m:ci><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply></m:apply><m:apply><m:times/><m:ci>k</m:ci><m:ci>S</m:ci><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">4</span>)</td>
	</tr>
      
      </table>
    <p class="p">with the initial values <m:math display="inline"><m:semantics><m:mrow><m:mrow><m:mi>s</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mn>1</m:mn><m:mo>,</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mi>S</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mn>1</m:mn><m:mo>,</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:apply><m:times/><m:ci>s</m:ci><m:apply><m:interval closure="open"/><m:cn>1</m:cn><m:cn>1</m:cn></m:apply></m:apply><m:eq/><m:apply><m:times/><m:ci>S</m:ci><m:apply><m:interval closure="open"/><m:cn>1</m:cn><m:cn>1</m:cn></m:apply></m:apply><m:eq/><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math>
and the “boundary conditions”
<m:math display="inline"><m:semantics><m:mrow><m:mrow><m:mi>s</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>i</m:mi><m:mo>,</m:mo><m:mi>j</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mi>S</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>i</m:mi><m:mo>,</m:mo><m:mi>j</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:apply><m:times/><m:ci>s</m:ci><m:apply><m:interval closure="open"/><m:ci>i</m:ci><m:ci>j</m:ci></m:apply></m:apply><m:eq/><m:apply><m:times/><m:ci>S</m:ci><m:apply><m:interval closure="open"/><m:ci>i</m:ci><m:ci>j</m:ci></m:apply></m:apply><m:eq/><m:cn>0</m:cn></m:apply></m:annotation-xml></m:semantics></m:math> for
<m:math display="inline"><m:semantics><m:mrow><m:mi>i</m:mi><m:mo>&lt;</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:lt/><m:ci>i</m:ci><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math>, <m:math display="inline"><m:semantics><m:mrow><m:mi>j</m:mi><m:mo>&lt;</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:lt/><m:ci>j</m:ci><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math> and for <m:math display="inline"><m:semantics><m:mrow><m:mi>i</m:mi><m:mo>&lt;</m:mo><m:mi>j</m:mi></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:lt/><m:ci>i</m:ci><m:ci>j</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>.
The combinatorial significance of the Stirling numbers has been amply
discussed <cite class="cite">[<a href="#bib.bib6" title="" class="ref">6</a>]</cite>.</p>
  </div>

  <div class="para" id="S2.p2">
    <p class="p">Several generalisations of the Stirling numbers appeared in the
mathematical literature [7-12].
In anticipation of further development we shall refer to them
generically as deformed Stirling numbers. In this context
we wish to distinguish between
the two widely used forms of “deformed numbers”
<m:math display="inline"><m:semantics><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>=</m:mo><m:mfrac><m:mrow><m:msup><m:mi>q</m:mi><m:mi>x</m:mi></m:msup><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow><m:mrow><m:mi>q</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfrac></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:divide/><m:apply><m:minus/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:ci>x</m:ci></m:apply><m:cn>1</m:cn></m:apply><m:apply><m:minus/><m:ci>q</m:ci><m:cn>1</m:cn></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>, the usual choice in the
mathematical literature on <m:math display="inline"><m:semantics><m:mi>q</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>q</m:ci></m:annotation-xml></m:semantics></m:math>-analysis <cite class="cite">[<a href="#bib.bib13" title="" class="ref">13</a>]</cite>, and
<m:math display="inline"><m:semantics><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>P</m:mi></m:msub><m:mo>=</m:mo><m:mfrac><m:mrow><m:msup><m:mi>q</m:mi><m:mi>x</m:mi></m:msup><m:mo>-</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mi>x</m:mi></m:mrow></m:msup></m:mrow><m:mrow><m:mi>q</m:mi><m:mo>-</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup></m:mrow></m:mfrac></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>P</m:ci></m:apply><m:apply><m:divide/><m:apply><m:minus/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:ci>x</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:ci>x</m:ci></m:apply></m:apply></m:apply><m:apply><m:minus/><m:ci>q</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:cn>1</m:cn></m:apply></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>,
which is common to the recent physical literature and to
the literature on quantum groups.
A generalisation was recently proposed by
Wachs and White <cite class="cite">[<a href="#bib.bib12" title="" class="ref">12</a>]</cite>, which can be written in the form
<m:math display="inline"><m:semantics><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>=</m:mo><m:mfrac><m:mrow><m:msup><m:mi>q</m:mi><m:mi>x</m:mi></m:msup><m:mo>-</m:mo><m:msup><m:mi>p</m:mi><m:mi>x</m:mi></m:msup></m:mrow><m:mrow><m:mi>q</m:mi><m:mo>-</m:mo><m:mi>p</m:mi></m:mrow></m:mfrac></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>G</m:ci></m:apply><m:apply><m:divide/><m:apply><m:minus/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:ci>x</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:ci>x</m:ci></m:apply></m:apply><m:apply><m:minus/><m:ci>q</m:ci><m:ci>p</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>.
This form contains <m:math display="inline"><m:semantics><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:annotation-xml encoding="MathML-Content"><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>M</m:ci></m:apply></m:annotation-xml></m:semantics></m:math> and <m:math display="inline"><m:semantics><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>P</m:mi></m:msub><m:annotation-xml encoding="MathML-Content"><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>P</m:ci></m:apply></m:annotation-xml></m:semantics></m:math> as special cases,
corresponding to the choices <m:math display="inline"><m:semantics><m:mrow><m:mi>p</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>p</m:ci><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math> and <m:math display="inline"><m:semantics><m:mrow><m:mi>p</m:mi><m:mo>=</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>p</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:cn>1</m:cn></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>, respectively.
We shall write <m:math display="inline"><m:semantics><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mrow><m:mi>M</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mi>q</m:mi></m:mfenced></m:mrow></m:msub><m:annotation-xml encoding="MathML-Content"><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:apply><m:times/><m:ci>M</m:ci><m:ci>q</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>, <m:math display="inline"><m:semantics><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mrow><m:mi>P</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mi>q</m:mi></m:mfenced></m:mrow></m:msub><m:annotation-xml encoding="MathML-Content"><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:apply><m:times/><m:ci>P</m:ci><m:ci>q</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> and
<m:math display="inline"><m:semantics><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mrow><m:mi>G</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>p</m:mi><m:mo>,</m:mo><m:mi>q</m:mi></m:mrow></m:mfenced></m:mrow></m:msub><m:annotation-xml encoding="MathML-Content"><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:apply><m:times/><m:ci>G</m:ci><m:apply><m:interval closure="open"/><m:ci>p</m:ci><m:ci>q</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> instead of the symbols introduced above
whenever the choice of the parameters <m:math display="inline"><m:semantics><m:mi>q</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>q</m:ci></m:annotation-xml></m:semantics></m:math> and/or <m:math display="inline"><m:semantics><m:mi>p</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>p</m:ci></m:annotation-xml></m:semantics></m:math> will
have to be explicated.
The identities</p>
  <table class="equation" id="S2.E5">
      
      
	<tr valign="baseline" class="equation" id="S2.E5">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mrow><m:mi>P</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mi>q</m:mi></m:mfenced></m:mrow></m:msub><m:mo>=</m:mo><m:mrow><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mi>x</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mrow><m:mi>M</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:msup><m:mi>q</m:mi><m:mn>2</m:mn></m:msup></m:mfenced></m:mrow></m:msub><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mrow><m:mi>G</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>p</m:mi><m:mo>,</m:mo><m:mi>q</m:mi></m:mrow></m:mfenced></m:mrow></m:msub></m:mrow><m:mo>=</m:mo><m:mrow><m:msup><m:mi>p</m:mi><m:mrow><m:mi>x</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mrow><m:mi>M</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>q</m:mi><m:mo>/</m:mo><m:mi>p</m:mi></m:mrow></m:mfenced></m:mrow></m:msub><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mrow><m:mi>G</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>p</m:mi><m:mo>,</m:mo><m:mi>q</m:mi></m:mrow></m:mfenced></m:mrow></m:msub></m:mrow><m:mo>=</m:mo><m:mrow><m:msup><m:mfenced open="(" close=")"><m:msqrt><m:mrow><m:mi>p</m:mi><m:mo>⁢</m:mo><m:mi>q</m:mi></m:mrow></m:msqrt></m:mfenced><m:mrow><m:mi>x</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mrow><m:mi>P</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:msqrt><m:mrow><m:mi>q</m:mi><m:mo>/</m:mo><m:mi>p</m:mi></m:mrow></m:msqrt></m:mfenced></m:mrow></m:msub></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:apply><m:times/><m:ci>P</m:ci><m:ci>q</m:ci></m:apply></m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:plus/><m:apply><m:minus/><m:ci>x</m:ci></m:apply><m:cn>1</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:apply><m:times/><m:ci>M</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:cn>2</m:cn></m:apply></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:apply><m:times/><m:ci>G</m:ci><m:apply><m:interval closure="open"/><m:ci>p</m:ci><m:ci>q</m:ci></m:apply></m:apply></m:apply></m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:minus/><m:ci>x</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:apply><m:times/><m:ci>M</m:ci><m:apply><m:divide/><m:ci>q</m:ci><m:ci>p</m:ci></m:apply></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:apply><m:times/><m:ci>G</m:ci><m:apply><m:interval closure="open"/><m:ci>p</m:ci><m:ci>q</m:ci></m:apply></m:apply></m:apply></m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:ci/><m:apply><m:times/><m:ci>p</m:ci><m:ci>q</m:ci></m:apply></m:apply><m:apply><m:minus/><m:ci>x</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:apply><m:times/><m:ci>P</m:ci><m:apply><m:ci/><m:apply><m:divide/><m:ci>q</m:ci><m:ci>p</m:ci></m:apply></m:apply></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">5</span>)</td>
	</tr>
      
      </table>
    <p class="p">illustrate the notation and exhibit
some of the elementary properties of these deformed numbers.</p>
  </div>

  <div class="para" id="S2.p3">
    <p class="p">One of the generalisations of the Stirling numbers <cite class="cite">[<a href="#bib.bib10" title="" class="ref">10</a>]</cite> involves
a descending product of M-type deformed
numbers expressed in terms of the powers
of the M-type deformed number <m:math display="inline"><m:semantics><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:annotation-xml encoding="MathML-Content"><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>M</m:ci></m:apply></m:annotation-xml></m:semantics></m:math></p>
  <table class="equation" id="S2.E6">
      
      
	<tr valign="baseline" class="equation" id="S2.E6">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>x</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>⁢</m:mo><m:mi mathvariant="normal">⋯</m:mi><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>x</m:mi><m:mo>-</m:mo><m:mi>k</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced><m:mi>M</m:mi></m:msub></m:mrow><m:mo>=</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>m</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>k</m:mi></m:mover><m:msub><m:mi>s</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msubsup><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>M</m:mi><m:mi>m</m:mi></m:msubsup></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:minus/><m:ci>x</m:ci><m:cn>1</m:cn></m:apply><m:ci>M</m:ci></m:apply><m:ci>⋯</m:ci><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:plus/><m:apply><m:minus/><m:ci>x</m:ci><m:ci>k</m:ci></m:apply><m:cn>1</m:cn></m:apply><m:ci>M</m:ci></m:apply></m:apply><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>s</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>M</m:ci></m:apply><m:ci>m</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">6</span>)</td>
	</tr>
      
      </table>
    <p class="p">and the corresponding inverse relation</p>
  <table class="equation" id="S2.E7">
      
      
	<tr valign="baseline" class="equation" id="S2.E7">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msubsup><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>M</m:mi><m:mi>m</m:mi></m:msubsup><m:mo>=</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>k</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>m</m:mi></m:mover><m:msub><m:mi>S</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>x</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>⁢</m:mo><m:mi mathvariant="normal">⋯</m:mi><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>x</m:mi><m:mo>-</m:mo><m:mi>k</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced><m:mi>M</m:mi></m:msub></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>M</m:ci></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:minus/><m:ci>x</m:ci><m:cn>1</m:cn></m:apply><m:ci>M</m:ci></m:apply><m:ci>⋯</m:ci><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:plus/><m:apply><m:minus/><m:ci>x</m:ci><m:ci>k</m:ci></m:apply><m:cn>1</m:cn></m:apply><m:ci>M</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">7</span>)</td>
	</tr>
      
      </table>
    <p class="p">Using the defining relations it is easy
to show that the deformed Stirling numbers <m:math display="inline"><m:semantics><m:mrow><m:msub><m:mi>s</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>s</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> and <m:math display="inline"><m:semantics><m:mrow><m:msub><m:mi>S</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>,
which are referred to in the mathematical literature as <m:math display="inline"><m:semantics><m:mi>q</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>q</m:ci></m:annotation-xml></m:semantics></m:math>-Stirling
numbers of the first and second kind, respectively,
satisfy the recurrence relations</p>
  <table class="equation" id="S2.E8">
      
      
	<tr valign="baseline" class="equation" id="S2.E8">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msub><m:mi>s</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:mi>k</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mi>k</m:mi></m:mrow></m:msup><m:mo>⁢</m:mo><m:none/><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:msub><m:mi>s</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mrow><m:mi>m</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mrow></m:mfenced></m:mrow><m:mo>-</m:mo><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>⁢</m:mo><m:msub><m:mi>s</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:none/></m:mrow></m:mrow></m:mfenced></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>s</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:apply><m:plus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply><m:ci>m</m:ci></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:ci>k</m:ci></m:apply></m:apply><m:ci/><m:apply><m:minus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>s</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:apply><m:minus/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>s</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply><m:ci/></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">8</span>)</td>
	</tr>
      
      </table>
    <p class="p">and</p>
  <table class="equation" id="S2.E9">
      
      
	<tr valign="baseline" class="equation" id="S2.E9">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:msub><m:mi>S</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:mi>m</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mrow><m:msup><m:mi>q</m:mi><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msub><m:mi>S</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mrow></m:mfenced></m:mrow><m:mo>+</m:mo><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>⁢</m:mo><m:msub><m:mi>S</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow></m:mrow></m:mrow><m:mo>,</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:apply><m:plus/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply><m:ci>k</m:ci></m:apply></m:apply><m:apply><m:plus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">9</span>)</td>
	</tr>
      
      </table>
    <p class="p">with “boundary conditions” and initial values identical with those
specified above for the conventional Stirling numbers.</p>
  </div>

  <div class="para" id="S2.p4">
    <p class="p">A slight modification in the form of the descending product, replacing
the factors <m:math display="inline"><m:semantics><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>x</m:mi><m:mo>-</m:mo><m:mi>i</m:mi></m:mrow></m:mfenced><m:mi>M</m:mi></m:msub><m:annotation-xml encoding="MathML-Content"><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:minus/><m:ci>x</m:ci><m:ci>i</m:ci></m:apply><m:ci>M</m:ci></m:apply></m:annotation-xml></m:semantics></m:math> by <m:math display="inline"><m:semantics><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>-</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>i</m:mi></m:mfenced><m:mi>M</m:mi></m:msub></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:minus/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>i</m:ci><m:ci>M</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>, results in the relations [8-10]</p>
  <table class="equation" id="S2.E10">
      
      
	<tr valign="baseline" class="equation" id="S2.E10">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>-</m:mo><m:msub><m:mfenced open="[" close="]"><m:mn>1</m:mn></m:mfenced><m:mi>M</m:mi></m:msub></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mi mathvariant="normal">⋯</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>-</m:mo><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced><m:mi>M</m:mi></m:msub></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>m</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>k</m:mi></m:mover><m:msub><m:mover accent="true"><m:mi>s</m:mi><m:mo>~</m:mo></m:mover><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msubsup><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>M</m:mi><m:mi>m</m:mi></m:msubsup></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:minus/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:cn>1</m:cn><m:ci>M</m:ci></m:apply></m:apply><m:ci>⋯</m:ci><m:apply><m:minus/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply><m:ci>M</m:ci></m:apply></m:apply></m:apply><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>~</m:ci><m:ci>s</m:ci></m:apply><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>M</m:ci></m:apply><m:ci>m</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">10</span>)</td>
	</tr>
      
      </table>
    <p class="p">and</p>
  <table class="equation" id="S2.E11">
      
      
	<tr valign="baseline" class="equation" id="S2.E11">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msubsup><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>M</m:mi><m:mi>m</m:mi></m:msubsup><m:mo>=</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>k</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>m</m:mi></m:mover><m:msub><m:mover accent="true"><m:mi>S</m:mi><m:mo>~</m:mo></m:mover><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>-</m:mo><m:msub><m:mfenced open="[" close="]"><m:mn>1</m:mn></m:mfenced><m:mi>M</m:mi></m:msub></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mi mathvariant="normal">⋯</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>x</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>-</m:mo><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced><m:mi>M</m:mi></m:msub></m:mrow></m:mfenced></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>M</m:ci></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>~</m:ci><m:ci>S</m:ci></m:apply><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:minus/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:cn>1</m:cn><m:ci>M</m:ci></m:apply></m:apply><m:ci>⋯</m:ci><m:apply><m:minus/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>x</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply><m:ci>M</m:ci></m:apply></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">11</span>)</td>
	</tr>
      
      </table>
    <p class="p">Starting with these defining relations and using the identity
<cite class="cite">[<a href="#bib.bib9" title="" class="ref">9</a>]</cite></p>
  <table class="equation" id="S2.E12">
      
      
	<tr valign="baseline" class="equation" id="S2.E12">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>a</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>-</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>b</m:mi></m:mfenced><m:mi>M</m:mi></m:msub></m:mrow><m:mo>=</m:mo><m:mrow><m:msup><m:mi>q</m:mi><m:mi>b</m:mi></m:msup><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>a</m:mi><m:mo>-</m:mo><m:mi>b</m:mi></m:mrow></m:mfenced><m:mi>M</m:mi></m:msub></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:minus/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>a</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>b</m:ci><m:ci>M</m:ci></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:ci>b</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:minus/><m:ci>a</m:ci><m:ci>b</m:ci></m:apply><m:ci>M</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">12</span>)</td>
	</tr>
      
      </table>
    <p class="p">we obtain the recurrence relations</p>
  <table class="equation" id="S2.E13">
      
      
	<tr valign="baseline" class="equation" id="S2.E13">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msub><m:mover accent="true"><m:mi>s</m:mi><m:mo>~</m:mo></m:mover><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:mi>k</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mrow><m:msub><m:mover accent="true"><m:mi>s</m:mi><m:mo>~</m:mo></m:mover><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mrow><m:mi>m</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mrow></m:mfenced></m:mrow><m:mo>-</m:mo><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>⁢</m:mo><m:msub><m:mover accent="true"><m:mi>s</m:mi><m:mo>~</m:mo></m:mover><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced></m:mrow></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>~</m:ci><m:ci>s</m:ci></m:apply><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:apply><m:plus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply><m:ci>m</m:ci></m:apply></m:apply><m:apply><m:minus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>~</m:ci><m:ci>s</m:ci></m:apply><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:apply><m:minus/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>~</m:ci><m:ci>s</m:ci></m:apply><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">13</span>)</td>
	</tr>
      
      </table>
    <p class="p">and</p>
  <table class="equation" id="S2.E14">
      
      
	<tr valign="baseline" class="equation" id="S2.E14">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:msub><m:mover accent="true"><m:mi>S</m:mi><m:mo>~</m:mo></m:mover><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:mi>m</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mrow><m:msub><m:mover accent="true"><m:mi>S</m:mi><m:mo>~</m:mo></m:mover><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mrow></m:mfenced></m:mrow><m:mo>+</m:mo><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>⁢</m:mo><m:msub><m:mover accent="true"><m:mi>S</m:mi><m:mo>~</m:mo></m:mover><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow></m:mrow></m:mrow><m:mo>,</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>~</m:ci><m:ci>S</m:ci></m:apply><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:apply><m:plus/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply><m:ci>k</m:ci></m:apply></m:apply><m:apply><m:plus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>~</m:ci><m:ci>S</m:ci></m:apply><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>~</m:ci><m:ci>S</m:ci></m:apply><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">14</span>)</td>
	</tr>
      
      </table>
    <p class="p">where the “boundary conditions” and initial values are, again,
as above. Note that</p>
  <table class="equation" id="S2.E15">
      
      
	<tr valign="baseline" class="equation" id="S2.E15">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:msub><m:mover accent="true"><m:mi>s</m:mi><m:mo>~</m:mo></m:mover><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msup><m:mi>q</m:mi><m:mrow><m:mrow><m:mi>k</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msub><m:mi>s</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msub><m:mover accent="true"><m:mi>S</m:mi><m:mo>~</m:mo></m:mover><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mrow><m:mrow><m:mi>k</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:mrow></m:msup><m:mo>⁢</m:mo><m:msub><m:mi>S</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>~</m:ci><m:ci>s</m:ci></m:apply><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply></m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:divide/><m:apply><m:times/><m:ci>k</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>s</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>~</m:ci><m:ci>S</m:ci></m:apply><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:apply><m:divide/><m:apply><m:times/><m:ci>k</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">15</span>)</td>
	</tr>
      
      </table>
    <p class="p">The two sets of deformed Stirling numbers of the first and second
kinds,
as well as the conventional Stirling numbers to which they
reduce in the limit <m:math display="inline"><m:semantics><m:mrow><m:mi>q</m:mi><m:mo>→</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>→</m:ci><m:ci>q</m:ci><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math>, satisfy the following
dual relations</p>
  <table class="equation" id="S2.E16">
      
      
	<tr valign="baseline" class="equation" id="S2.E16">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>m</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>k</m:mi></m:mover><m:msub><m:mi>s</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msub><m:mi>S</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:msup><m:mi>k</m:mi><m:mo>′</m:mo></m:msup></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mi>δ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:msup><m:mi>k</m:mi><m:mo>′</m:mo></m:msup></m:mrow></m:mfenced></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>s</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>k</m:ci><m:ci>′</m:ci></m:apply></m:apply></m:apply></m:apply><m:apply><m:times/><m:ci>δ</m:ci><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>k</m:ci><m:ci>′</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">16</span>)</td>
	</tr>
      
      </table>
    <p class="p">and</p>
  <table class="equation" id="S2.E17">
      
      
	<tr valign="baseline" class="equation" id="S2.E17">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>k</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>m</m:mi></m:mover><m:msub><m:mi>S</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msub><m:mi>s</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:msup><m:mi>m</m:mi><m:mo>′</m:mo></m:msup></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mi>δ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:msup><m:mi>m</m:mi><m:mo>′</m:mo></m:msup></m:mrow></m:mfenced></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>s</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>m</m:ci><m:ci>′</m:ci></m:apply></m:apply></m:apply></m:apply><m:apply><m:times/><m:ci>δ</m:ci><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>m</m:ci><m:ci>′</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">17</span>)</td>
	</tr>
      
      </table>
    <p class="p">An additional set of deformed Stirling numbers of the second kind
was recently introduced by Wachs and White <cite class="cite">[<a href="#bib.bib12" title="" class="ref">12</a>]</cite>.
Their definition is motivated by combinatorial
considerations and has no algebraic origin. Their recurrence
relation reads</p>
  <table class="equation" id="S2.E18">
      
      
	<tr valign="baseline" class="equation" id="S2.E18">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msub><m:mi>S</m:mi><m:mrow><m:mi>p</m:mi><m:mo>,</m:mo><m:mi>q</m:mi></m:mrow></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:mi>m</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mrow><m:msup><m:mi>p</m:mi><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msub><m:mi>S</m:mi><m:mrow><m:mi>p</m:mi><m:mo>,</m:mo><m:mi>q</m:mi></m:mrow></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mrow></m:mfenced></m:mrow><m:mo>+</m:mo><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:msub><m:mi>S</m:mi><m:mrow><m:mi>p</m:mi><m:mo>,</m:mo><m:mi>q</m:mi></m:mrow></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:apply><m:list/><m:ci>p</m:ci><m:ci>q</m:ci></m:apply></m:apply><m:apply><m:interval closure="open"/><m:apply><m:plus/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply><m:ci>k</m:ci></m:apply></m:apply><m:apply><m:plus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:apply><m:list/><m:ci>p</m:ci><m:ci>q</m:ci></m:apply></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>G</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:apply><m:list/><m:ci>p</m:ci><m:ci>q</m:ci></m:apply></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">18</span>)</td>
	</tr>
      
      </table>
    <p class="p">and it reduces to (<a href="#S2.E14" title="Eq.14 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">14</a>) for <m:math display="inline"><m:semantics><m:mrow><m:mi>p</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>p</m:ci><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math>.</p>
  </div>

    </div>
  
    <div class="section" id="S3">
    <h2 class="title section-title"><span class="refnum"><span class="reftype">§ </span>3. </span>Some algebraic properties of deformed boson operators</h2>
  <div class="para" id="S3.p1">
    <p class="p">In the context of recent interest in quantum groups and their
realization, three types of
deformed boson operators have
been introduced [3,4,14]. The most straightforward definition starts by
postulating a Fock space on which creation (<m:math display="inline"><m:semantics><m:mi>a</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>a</m:ci></m:annotation-xml></m:semantics></m:math>), annihilation
(<m:math display="inline"><m:semantics><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup><m:annotation-xml encoding="MathML-Content"><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>)
and number (<m:math display="inline"><m:semantics><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>) operators are
defined in analogy with the conventional boson operators. The general
form postulated is</p>
  <table class="equation" id="S3.E19">
      
      
	<tr valign="baseline" class="equation" id="S3.E19">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mi>a</m:mi><m:mo>|</m:mo><m:mi>l</m:mi><m:mo>&gt;</m:mo><m:mo>=</m:mo><m:msqrt><m:mfenced open="[" close="]"><m:mi>l</m:mi></m:mfenced></m:msqrt><m:mo>|</m:mo><m:mi>l</m:mi><m:mo>-</m:mo><m:mn>1</m:mn><m:mo>&gt;</m:mo><m:mi>a</m:mi><m:msup><m:mi/><m:mo>†</m:mo></m:msup><m:mo>|</m:mo><m:mi>l</m:mi><m:mo>&gt;</m:mo><m:mo>=</m:mo><m:msqrt><m:mfenced open="[" close="]"><m:mrow><m:mi>l</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:msqrt><m:mo>|</m:mo><m:mi>l</m:mi><m:mo>+</m:mo><m:mn>1</m:mn><m:mo>&gt;</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>|</m:mo><m:mi>l</m:mi><m:mo>≥</m:mo><m:mi>l</m:mi><m:mo>|</m:mo><m:mi>l</m:mi><m:mo>&gt;</m:mo><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:ci>a</m:ci></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">19</span>)</td>
	</tr>
      
      </table>
    <p class="p">It follows immediately that <m:math display="inline"><m:semantics><m:mrow><m:mrow><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup><m:mo>⁢</m:mo><m:mi>a</m:mi></m:mrow><m:mo>=</m:mo><m:mfenced open="[" close="]"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>a</m:ci></m:apply><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> and
<m:math display="inline"><m:semantics><m:mrow><m:mrow><m:mi>a</m:mi><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mrow><m:mo>=</m:mo><m:mfenced open="[" close="]"><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:ci>a</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply></m:apply><m:apply><m:plus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:cn>1</m:cn></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>.
The two widely used forms of the deformed
bosons are obtained by choosing either
<m:math display="inline"><m:semantics><m:mrow><m:mfenced open="[" close="]"><m:mi>l</m:mi></m:mfenced><m:mo>=</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>l</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>=</m:mo><m:mfrac><m:mrow><m:msup><m:mi>q</m:mi><m:mi>l</m:mi></m:msup><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow><m:mrow><m:mi>q</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfrac></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:ci>l</m:ci><m:eq/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>l</m:ci><m:ci>M</m:ci></m:apply><m:eq/><m:apply><m:divide/><m:apply><m:minus/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:ci>l</m:ci></m:apply><m:cn>1</m:cn></m:apply><m:apply><m:minus/><m:ci>q</m:ci><m:cn>1</m:cn></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> or
<m:math display="inline"><m:semantics><m:mrow><m:mfenced open="[" close="]"><m:mi>l</m:mi></m:mfenced><m:mo>=</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>l</m:mi></m:mfenced><m:mi>P</m:mi></m:msub><m:mo>=</m:mo><m:mfrac><m:mrow><m:msup><m:mi>q</m:mi><m:mi>l</m:mi></m:msup><m:mo>-</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mi>l</m:mi></m:mrow></m:msup></m:mrow><m:mrow><m:mi>q</m:mi><m:mo>-</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup></m:mrow></m:mfrac></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:ci>l</m:ci><m:eq/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>l</m:ci><m:ci>P</m:ci></m:apply><m:eq/><m:apply><m:divide/><m:apply><m:minus/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:ci>l</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:ci>l</m:ci></m:apply></m:apply></m:apply><m:apply><m:minus/><m:ci>q</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:cn>1</m:cn></m:apply></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>.
A generalisation was recently proposed by
Chakrabarti and Jagannathan <cite class="cite">[<a href="#bib.bib14" title="" class="ref">14</a>]</cite>. We shall adhere to
the notation introduced by Wachs and White <cite class="cite">[<a href="#bib.bib12" title="" class="ref">12</a>]</cite> and
write this generalisation in the form
<m:math display="inline"><m:semantics><m:mrow><m:mfenced open="[" close="]"><m:mi>l</m:mi></m:mfenced><m:mo>=</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>l</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>=</m:mo><m:mfrac><m:mrow><m:msup><m:mi>q</m:mi><m:mi>l</m:mi></m:msup><m:mo>-</m:mo><m:msup><m:mi>p</m:mi><m:mi>l</m:mi></m:msup></m:mrow><m:mrow><m:mi>q</m:mi><m:mo>-</m:mo><m:mi>p</m:mi></m:mrow></m:mfrac></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:ci>l</m:ci><m:eq/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>l</m:ci><m:ci>G</m:ci></m:apply><m:eq/><m:apply><m:divide/><m:apply><m:minus/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:ci>l</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:ci>l</m:ci></m:apply></m:apply><m:apply><m:minus/><m:ci>q</m:ci><m:ci>p</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>, which is
trivially modified relative to that introduced in
Ref. <cite class="cite">[<a href="#bib.bib14" title="" class="ref">14</a>]</cite>.
As a consequence of a remark made in the previous section, this
third type of deformed boson contains the first two as special cases.</p>
  </div>

  <div class="para" id="S3.p2">
    <p class="p">The deformed bosons as defined by Eq. (<a href="#S3.E19" title="Eq.19 in §3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">19</a>) are not associated
with any a priori specification of a (possibly deformed)
commutation relation. Choosing a parameter <m:math display="inline"><m:semantics><m:mi>Q</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>Q</m:ci></m:annotation-xml></m:semantics></m:math>, which does not have to
be related to the two parameters <m:math display="inline"><m:semantics><m:mi>p</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>p</m:ci></m:annotation-xml></m:semantics></m:math> and <m:math display="inline"><m:semantics><m:mi>q</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>q</m:ci></m:annotation-xml></m:semantics></m:math> so far introduced,
the deformed bosons are found to satisfy the deformed commutation
relation</p>
  <table class="equation" id="S3.E20">
      
      
	<tr valign="baseline" class="equation" id="S3.E20">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>a</m:mi><m:mo>,</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mrow></m:mfenced><m:mi>Q</m:mi></m:msub><m:mo>=</m:mo><m:mrow><m:mrow><m:mi>a</m:mi><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mrow><m:mo>-</m:mo><m:mrow><m:mi>Q</m:mi><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup><m:mo>⁢</m:mo><m:mi>a</m:mi></m:mrow></m:mrow><m:mo>=</m:mo><m:mrow><m:mi>ϕ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mfrac><m:mn>1</m:mn><m:mrow><m:mi>q</m:mi><m:mo>-</m:mo><m:mi>p</m:mi></m:mrow></m:mfrac><m:mo>⁢</m:mo><m:none/><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:msup><m:mi>q</m:mi><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:msup><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>q</m:mi><m:mo>-</m:mo><m:mi>Q</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>+</m:mo><m:mrow><m:msup><m:mi>p</m:mi><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:msup><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>Q</m:mi><m:mo>-</m:mo><m:mi>p</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:none/></m:mrow></m:mrow></m:mfenced></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:interval closure="closed"/><m:ci>a</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply></m:apply><m:ci>Q</m:ci></m:apply><m:eq/><m:apply><m:minus/><m:apply><m:times/><m:ci>a</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply></m:apply><m:apply><m:times/><m:ci>Q</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>a</m:ci></m:apply></m:apply><m:eq/><m:apply><m:times/><m:ci>ϕ</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:eq/><m:apply><m:times/><m:apply><m:divide/><m:cn>1</m:cn><m:apply><m:minus/><m:ci>q</m:ci><m:ci>p</m:ci></m:apply></m:apply><m:ci/><m:apply><m:plus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:apply><m:minus/><m:ci>q</m:ci><m:ci>Q</m:ci></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:apply><m:minus/><m:ci>Q</m:ci><m:ci>p</m:ci></m:apply><m:ci/></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">20</span>)</td>
	</tr>
      
      </table>
    <p class="p">Since the choice of <m:math display="inline"><m:semantics><m:mi>Q</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>Q</m:ci></m:annotation-xml></m:semantics></m:math> is arbitrary we can opt to be guided by the
requirement that the form of <m:math display="inline"><m:semantics><m:mrow><m:mi>ϕ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:ci>ϕ</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> be as simple as possible
or by some other relevant criterion.
The conventional choice <m:math display="inline"><m:semantics><m:mrow><m:mi>Q</m:mi><m:mo>=</m:mo><m:mi>q</m:mi></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>Q</m:ci><m:ci>q</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>, to which we will eventually adhere,
results in</p>
  <table class="equation" id="S3.E21">
      
      
	<tr valign="baseline" class="equation" id="S3.E21">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:mrow><m:mi>a</m:mi><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mrow><m:mo>-</m:mo><m:mrow><m:mi>q</m:mi><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup><m:mo>⁢</m:mo><m:mi>a</m:mi></m:mrow></m:mrow><m:mo>=</m:mo><m:mrow><m:msub><m:mi>ϕ</m:mi><m:mi>M</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced></m:mrow><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:mo>,</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:apply><m:minus/><m:apply><m:times/><m:ci>a</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply></m:apply><m:apply><m:times/><m:ci>q</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>a</m:ci></m:apply></m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>ϕ</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:eq/><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">21</span>)</td>
	</tr>
      
      </table><table class="equation" id="S3.E22">
      
      
	<tr valign="baseline" class="equation" id="S3.E22">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:mi>a</m:mi><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mrow><m:mo>-</m:mo><m:mrow><m:mi>q</m:mi><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup><m:mo>⁢</m:mo><m:mi>a</m:mi></m:mrow></m:mrow><m:mo>=</m:mo><m:mrow><m:msub><m:mi>ϕ</m:mi><m:mi>P</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced></m:mrow><m:mo>=</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:msup></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:apply><m:minus/><m:apply><m:times/><m:ci>a</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply></m:apply><m:apply><m:times/><m:ci>q</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>a</m:ci></m:apply></m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>ϕ</m:ci><m:ci>P</m:ci></m:apply><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">22</span>)</td>
	</tr>
      
      </table>
    <p class="p">and</p>
  <table class="equation" id="S3.E23">
      
      
	<tr valign="baseline" class="equation" id="S3.E23">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:mi>a</m:mi><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mrow><m:mo>-</m:mo><m:mrow><m:mi>q</m:mi><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup><m:mo>⁢</m:mo><m:mi>a</m:mi></m:mrow></m:mrow><m:mo>=</m:mo><m:mrow><m:msub><m:mi>ϕ</m:mi><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced></m:mrow><m:mo>=</m:mo><m:msup><m:mi>p</m:mi><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:msup></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:apply><m:minus/><m:apply><m:times/><m:ci>a</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply></m:apply><m:apply><m:times/><m:ci>q</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>a</m:ci></m:apply></m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>ϕ</m:ci><m:ci>G</m:ci></m:apply><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">23</span>)</td>
	</tr>
      
      </table>
    <p class="p">for the M-type, P-type and G-type bosons, respectively. We do not
label the creation and annihilation operators by indices such
as M, P or G because the nature of these operators is always
obvious from the context.
The choice <m:math display="inline"><m:semantics><m:mrow><m:mi>Q</m:mi><m:mo>=</m:mo><m:mi>p</m:mi></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>Q</m:ci><m:ci>p</m:ci></m:apply></m:annotation-xml></m:semantics></m:math> results in
<m:math display="inline"><m:semantics><m:mrow><m:mrow><m:mi>ϕ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced></m:mrow><m:mo>=</m:mo><m:msup><m:mi>q</m:mi><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:msup></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:ci>ϕ</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> for all the three cases.
For the M-type bosons (<m:math display="inline"><m:semantics><m:mrow><m:mi>p</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>p</m:ci><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math>) this choice implies <m:math display="inline"><m:semantics><m:mrow><m:mi>Q</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>Q</m:ci><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math>, i.e.,
the deformed commutation relation becomes
<m:math display="inline"><m:semantics><m:mrow><m:mrow><m:mrow><m:mi>a</m:mi><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mrow><m:mo>-</m:mo><m:mrow><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup><m:mo>⁢</m:mo><m:mi>a</m:mi></m:mrow></m:mrow><m:mo>=</m:mo><m:msup><m:mi>q</m:mi><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:msup></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:minus/><m:apply><m:times/><m:ci>a</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>a</m:ci></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>.
For the P-type bosons (<m:math display="inline"><m:semantics><m:mrow><m:mi>p</m:mi><m:mo>=</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>p</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:cn>1</m:cn></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>) this choice is the familiar
alternative to Eq. (<a href="#S3.E22" title="Eq.22 in §3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">22</a>), namely
<m:math display="inline"><m:semantics><m:mrow><m:mrow><m:mrow><m:mi>a</m:mi><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mrow><m:mo>-</m:mo><m:mrow><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup><m:mo>⁢</m:mo><m:mi>a</m:mi></m:mrow></m:mrow><m:mo>=</m:mo><m:msup><m:mi>q</m:mi><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:msup></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:minus/><m:apply><m:times/><m:ci>a</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:cn>1</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>a</m:ci></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>.
In a recent study of the extension of the
Campbell-Baker-Hausdorff formula to deformed bosons <cite class="cite">[<a href="#bib.bib16" title="" class="ref">16</a>]</cite>, it was
noted that the choice <m:math display="inline"><m:semantics><m:mrow><m:mi>Q</m:mi><m:mo>=</m:mo><m:mi>q</m:mi></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>Q</m:ci><m:ci>q</m:ci></m:apply></m:annotation-xml></m:semantics></m:math> is the most suitable one for
the M-type bosons, but that <m:math display="inline"><m:semantics><m:mrow><m:mi>Q</m:mi><m:mo>=</m:mo><m:mrow><m:mi>q</m:mi><m:mo>+</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>Q</m:ci><m:apply><m:minus/><m:apply><m:plus/><m:ci>q</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:cn>1</m:cn></m:apply></m:apply></m:apply><m:cn>1</m:cn></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> seems to have some advantages
for the P-type bosons. From the same point of view, one would
choose <m:math display="inline"><m:semantics><m:mrow><m:mi>Q</m:mi><m:mo>=</m:mo><m:mrow><m:mi>q</m:mi><m:mo>+</m:mo><m:mi>p</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>Q</m:ci><m:apply><m:minus/><m:apply><m:plus/><m:ci>q</m:ci><m:ci>p</m:ci></m:apply><m:cn>1</m:cn></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> for the G-type bosons.</p>
  </div>

  <div class="para" id="S3.p3">
    <p class="p">We shall also need the relation</p>
  <table class="equation" id="S3.E24">
      
      
	<tr valign="baseline" class="equation" id="S3.E24">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mrow><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup><m:mo>,</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mrow></m:mfenced><m:msup><m:mi>Q</m:mi><m:mi>k</m:mi></m:msup></m:msub><m:mo>=</m:mo><m:mrow><m:mi mathvariant="normal">Φ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:interval closure="closed"/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>Q</m:ci><m:ci>k</m:ci></m:apply></m:apply><m:apply><m:times/><m:ci>Φ</m:ci><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">24</span>)</td>
	</tr>
      
      </table>
    <p class="p">which can be viewed as an extension of Eq. (<a href="#S3.E20" title="Eq.20 in §3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">20</a>) in the sense
that <m:math display="inline"><m:semantics><m:mrow><m:mrow><m:mi mathvariant="normal">Φ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mn>1</m:mn><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mi>ϕ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:ci>Φ</m:ci><m:apply><m:interval closure="open"/><m:cn>1</m:cn><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply><m:apply><m:times/><m:ci>ϕ</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>.
One easily finds that</p>
  <table class="equation" id="S3.E25">
      
      
	<tr valign="baseline" class="equation" id="S3.E25">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:mi mathvariant="normal">Φ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mfrac><m:mn>1</m:mn><m:mrow><m:mi>q</m:mi><m:mo>-</m:mo><m:mi>p</m:mi></m:mrow></m:mfrac><m:mo>⁢</m:mo><m:none/><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:msup><m:mi>q</m:mi><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:msup><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>q</m:mi><m:mo>-</m:mo><m:mi>Q</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mrow><m:mi>G</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>Q</m:mi><m:mo>,</m:mo><m:mi>q</m:mi></m:mrow></m:mfenced></m:mrow></m:msub></m:mrow><m:mo>+</m:mo><m:mrow><m:msup><m:mi>p</m:mi><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:msup><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>Q</m:mi><m:mo>-</m:mo><m:mi>p</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mrow><m:mi>G</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>Q</m:mi><m:mo>,</m:mo><m:mi>p</m:mi></m:mrow></m:mfenced></m:mrow></m:msub><m:mo>⁢</m:mo><m:none/></m:mrow></m:mrow></m:mfenced></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:ci>Φ</m:ci><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:divide/><m:cn>1</m:cn><m:apply><m:minus/><m:ci>q</m:ci><m:ci>p</m:ci></m:apply></m:apply><m:ci/><m:apply><m:plus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:apply><m:minus/><m:ci>q</m:ci><m:ci>Q</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:apply><m:times/><m:ci>G</m:ci><m:apply><m:interval closure="open"/><m:ci>Q</m:ci><m:ci>q</m:ci></m:apply></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:apply><m:minus/><m:ci>Q</m:ci><m:ci>p</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:apply><m:times/><m:ci>G</m:ci><m:apply><m:interval closure="open"/><m:ci>Q</m:ci><m:ci>p</m:ci></m:apply></m:apply></m:apply><m:ci/></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">25</span>)</td>
	</tr>
      
      </table>
    <p class="p">We shall retain the conventional choice <m:math display="inline"><m:semantics><m:mrow><m:mi>Q</m:mi><m:mo>=</m:mo><m:mi>q</m:mi></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>Q</m:ci><m:ci>q</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>
for the three cases specified above.
With this choice we get</p>
  <table class="equation" id="S3.E26">
      
      
	<tr valign="baseline" class="equation" id="S3.E26">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:msub><m:mi mathvariant="normal">Φ</m:mi><m:mi>M</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>⁢</m:mo><m:msub><m:mi mathvariant="normal">Φ</m:mi><m:mi>P</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>P</m:mi></m:msub><m:mo>⁢</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:msup><m:mo>⁢</m:mo><m:msub><m:mi mathvariant="normal">Φ</m:mi><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:msup><m:mi>p</m:mi><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:msup></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>Φ</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>Φ</m:ci><m:ci>P</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>P</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>Φ</m:ci><m:ci>G</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>G</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">26</span>)</td>
	</tr>
      
      </table></div>

    </div>
  
    <div class="section" id="S4">
    <h2 class="title section-title"><span class="refnum"><span class="reftype">§ </span>4. </span>Normal ordering of powers of the deformed number operator</h2>
  <div class="para" id="S4.p1">
    <p class="p">The relevance of the ordinary Stirling numbers to the normal ordering of
powers of the boson number operator was demonstrated in
Ref. <cite class="cite">[<a href="#bib.bib15" title="" class="ref">15</a>]</cite>.
In the present section we consider some normal ordering properties
of the deformed bosons specified by the parameter choice <m:math display="inline"><m:semantics><m:mrow><m:mi>p</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>p</m:ci><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math> and
<m:math display="inline"><m:semantics><m:mrow><m:mi>Q</m:mi><m:mo>=</m:mo><m:mi>q</m:mi></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>Q</m:ci><m:ci>q</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>, which corresponds to the M-type boson operators
and to the deformed commutation relation (<a href="#S3.E21" title="Eq.21 in §3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">21</a>).
Up to a trivial interchange of <m:math display="inline"><m:semantics><m:mi>p</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>p</m:ci></m:annotation-xml></m:semantics></m:math> and <m:math display="inline"><m:semantics><m:mi>q</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>q</m:ci></m:annotation-xml></m:semantics></m:math>
this is the only combination of parameters for which the
deformed commutator does not depend on <m:math display="inline"><m:semantics><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>.
The other types of deformed boson operators are considered in the
following
section where it is found that they differ in a significant respect from
the case presently considered.</p>
  </div>

  <div class="para" id="S4.p2">
    <p class="p">In order to express an integral power of <m:math display="inline"><m:semantics><m:msub><m:mfenced open="[" close="]"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced><m:mi>M</m:mi></m:msub><m:annotation-xml encoding="MathML-Content"><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>M</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>
in a normally ordered form we can either formally write
such an expansion and obtain a recurrence relation for
the coefficients by applying Eq. (<a href="#S3.E21" title="Eq.21 in §3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">21</a>) or
use the deformed Stirling
numbers of the second kind directly. We shall present
both approaches because of the intrinsic interest of each one of them.</p>
  </div>

  <div class="para" id="S4.p3">
    <p class="p">In the direct approach, we start from the expansion</p>
  <table class="equation" id="S4.E27">
      
      
	<tr valign="baseline" class="equation" id="S4.E27">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msubsup><m:mfenced open="[" close="]"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced><m:mi>M</m:mi><m:mi>m</m:mi></m:msubsup><m:mo>=</m:mo><m:msup><m:mfenced open="(" close=")"><m:mrow><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup><m:mo>⁢</m:mo><m:mi>a</m:mi></m:mrow></m:mfenced><m:mi>m</m:mi></m:msup><m:mo>=</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>k</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>m</m:mi></m:mover><m:mi>c</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>M</m:ci></m:apply><m:ci>m</m:ci></m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>a</m:ci></m:apply><m:ci>m</m:ci></m:apply><m:eq/><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:times/><m:ci>c</m:ci><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">27</span>)</td>
	</tr>
      
      </table>
    <p class="p">Expressing <m:math display="inline"><m:semantics><m:msup><m:mfenced open="(" close=")"><m:mrow><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup><m:mo>⁢</m:mo><m:mi>a</m:mi></m:mrow></m:mfenced><m:mrow><m:mi>m</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:msup><m:annotation-xml encoding="MathML-Content"><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>a</m:ci></m:apply><m:apply><m:plus/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> by means of Eq. (<a href="#S4.E27" title="Eq.27 in §4. Normal ordering of powers of the deformed number operator in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">27</a>) and
using Eq. (<a href="#S3.E21" title="Eq.21 in §3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">21</a>), we obtain
a recurrence relation
which is identical with the one satisfied by <m:math display="inline"><m:semantics><m:mrow><m:msub><m:mi>S</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>,
Eq. (<a href="#S2.E9" title="Eq.9 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">9</a>). Moreover,
it is obvious from the defining equation (<a href="#S4.E27" title="Eq.27 in §4. Normal ordering of powers of the deformed number operator in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">27</a>) that
<m:math display="inline"><m:semantics><m:mrow><m:mrow><m:mi>c</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mn>1</m:mn><m:mo>,</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msub><m:mi>S</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mn>1</m:mn><m:mo>,</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:apply><m:times/><m:ci>c</m:ci><m:apply><m:interval closure="open"/><m:cn>1</m:cn><m:cn>1</m:cn></m:apply></m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:cn>1</m:cn><m:cn>1</m:cn></m:apply></m:apply><m:eq/><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math>. Thus, <m:math display="inline"><m:semantics><m:mrow><m:mrow><m:mi>c</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msub><m:mi>S</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:ci>c</m:ci><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>.</p>
  </div>

  <div class="para" id="S4.p4">
    <p class="p">A different derivation can be obtained by using the identity</p>
  <table class="equation" id="S4.E28">
      
      
	<tr valign="baseline" class="equation" id="S4.E28">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∏</m:mo><m:mrow><m:mi>i</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>0</m:mn></m:mrow></m:munder><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mover><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>-</m:mo><m:mi>i</m:mi></m:mrow></m:mfenced><m:mi>M</m:mi></m:msub></m:mrow><m:mo>=</m:mo><m:mrow><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>∏</m:ci><m:apply><m:eq/><m:ci>i</m:ci><m:cn>0</m:cn></m:apply></m:apply><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:minus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>i</m:ci></m:apply><m:ci>M</m:ci></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">28</span>)</td>
	</tr>
      
      </table>
    <p class="p">This identity follows by noting that application of both sides of
Eq. (<a href="#S4.E28" title="Eq.28 in §4. Normal ordering of powers of the deformed number operator in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">28</a>) on any member of the complete set
<m:math display="inline"><m:semantics><m:mfenced open="{" close="}"><m:mrow><m:mrow><m:mrow><m:mfenced open="|" close="&gt;"><m:mi>l</m:mi></m:mfenced><m:mo>;</m:mo><m:mi>l</m:mi></m:mrow><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mo>,</m:mo><m:mrow><m:mn> 1</m:mn><m:mo>,</m:mo><m:mi mathvariant="normal">⋯</m:mi></m:mrow></m:mrow></m:mfenced><m:annotation-xml encoding="MathML-Content"><m:apply><m:set/><m:apply><m:ci/><m:apply><m:eq/><m:apply><m:list/><m:apply><m:ci/><m:ci>l</m:ci></m:apply><m:ci>l</m:ci></m:apply><m:cn>0</m:cn></m:apply><m:apply><m:list/><m:cn> 1</m:cn><m:ci>⋯</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>
of eigenstates of the number operator
results in <m:math display="inline"><m:semantics><m:mrow><m:msubsup><m:mo>∏</m:mo><m:mrow><m:mi>i</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msubsup><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>l</m:mi><m:mo>-</m:mo><m:mi>i</m:mi></m:mrow></m:mfenced><m:mi>M</m:mi></m:msub></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>∏</m:ci><m:apply><m:eq/><m:ci>i</m:ci><m:cn>0</m:cn></m:apply></m:apply><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:minus/><m:ci>l</m:ci><m:ci>i</m:ci></m:apply><m:ci>M</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>.
Using Eq. (<a href="#S2.E7" title="Eq.7 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">7</a>) we obtain</p>
  <table class="equation" id="S4.E29">
      
      
	<tr valign="baseline" class="equation" id="S4.E29">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:msubsup><m:mfenced open="[" close="]"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced><m:mi>M</m:mi><m:mi>m</m:mi></m:msubsup><m:mo>=</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>k</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>m</m:mi></m:mover><m:msub><m:mi>S</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∏</m:mo><m:mrow><m:mi>i</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>0</m:mn></m:mrow></m:munder><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mover><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>-</m:mo><m:mi>i</m:mi></m:mrow></m:mfenced><m:mi>M</m:mi></m:msub></m:mrow></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>M</m:ci></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>∏</m:ci><m:apply><m:eq/><m:ci>i</m:ci><m:cn>0</m:cn></m:apply></m:apply><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:minus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>i</m:ci></m:apply><m:ci>M</m:ci></m:apply></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">29</span>)</td>
	</tr>
      
      </table>
    <p class="p">and substituting Eq. (<a href="#S4.E28" title="Eq.28 in §4. Normal ordering of powers of the deformed number operator in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">28</a>) we get the desired normally ordered
expansion</p>
  <table class="equation" id="S4.E30">
      
      
	<tr valign="baseline" class="equation" id="S4.E30">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msubsup><m:mfenced open="[" close="]"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced><m:mi>M</m:mi><m:mi>m</m:mi></m:msubsup><m:mo>=</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>k</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>m</m:mi></m:mover><m:msub><m:mi>S</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>M</m:ci></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">30</span>)</td>
	</tr>
      
      </table>
    <p class="p">We note in passing that an equivalent expansion could have been obtained
starting from the identity</p>
  <table class="equation" id="S4.E31">
      
      
	<tr valign="baseline" class="equation" id="S4.E31">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∏</m:mo><m:mrow><m:mi>i</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>0</m:mn></m:mrow></m:munder><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mover><m:mfenced open="(" close=")"><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>-</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>i</m:mi></m:mfenced><m:mi>M</m:mi></m:msub></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msup><m:mi>q</m:mi><m:mrow><m:mrow><m:mi>k</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>∏</m:ci><m:apply><m:eq/><m:ci>i</m:ci><m:cn>0</m:cn></m:apply></m:apply><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:apply><m:minus/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>M</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>i</m:ci><m:ci>M</m:ci></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:divide/><m:apply><m:times/><m:ci>k</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">31</span>)</td>
	</tr>
      
      </table>
    <p class="p">This identity can be proved either
by induction or by considering the effect
of both sides on the complete set of eigenstates of the number operator.
Using (<a href="#S2.E11" title="Eq.11 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">11</a>) and (<a href="#S4.E31" title="Eq.31 in §4. Normal ordering of powers of the deformed number operator in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">31</a>), we obtain the normally ordered
expansion of <m:math display="inline"><m:semantics><m:msubsup><m:mfenced open="[" close="]"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced><m:mi>M</m:mi><m:mi>m</m:mi></m:msubsup><m:annotation-xml encoding="MathML-Content"><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>M</m:ci></m:apply><m:ci>m</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>
in the form</p>
  <table class="equation" id="S4.E32">
      
      
	<tr valign="baseline" class="equation" id="S4.E32">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:msubsup><m:mfenced open="[" close="]"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced><m:mi>M</m:mi><m:mi>m</m:mi></m:msubsup><m:mo>=</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>k</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>m</m:mi></m:mover><m:msub><m:mover accent="true"><m:mi>S</m:mi><m:mo>~</m:mo></m:mover><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mrow><m:mi>k</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>M</m:ci></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>~</m:ci><m:ci>S</m:ci></m:apply><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:divide/><m:apply><m:times/><m:ci>k</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">32</span>)</td>
	</tr>
      
      </table>
    <p class="p">which is related to (<a href="#S4.E30" title="Eq.30 in §4. Normal ordering of powers of the deformed number operator in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">30</a>) by Eq. (<a href="#S2.E15" title="Eq.15 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">15</a>).</p>
  </div>

  <div class="para" id="S4.p5">
    <p class="p">In order  to obtain the inverse relation,
expressing a normally ordered product
as a function of the number operator, we note that
Eqs. (<a href="#S2.E6" title="Eq.6 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">6</a>) and (<a href="#S4.E28" title="Eq.28 in §4. Normal ordering of powers of the deformed number operator in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">28</a>) lead to</p>
  <table class="equation" id="S4.E33">
      
      
	<tr valign="baseline" class="equation" id="S4.E33">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow><m:mo>=</m:mo><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced><m:mi>M</m:mi></m:msub><m:mo>⁢</m:mo><m:mi mathvariant="normal">⋯</m:mi><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>-</m:mo><m:mi>k</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced><m:mi>M</m:mi></m:msub></m:mrow><m:mo>=</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>m</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>k</m:mi></m:mover><m:msub><m:mi>s</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msubsup><m:mfenced open="[" close="]"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced><m:mi>M</m:mi><m:mi>m</m:mi></m:msubsup></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>M</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:minus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:cn>1</m:cn></m:apply><m:ci>M</m:ci></m:apply><m:ci>⋯</m:ci><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:plus/><m:apply><m:minus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:cn>1</m:cn></m:apply><m:ci>M</m:ci></m:apply></m:apply><m:eq/><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>s</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>M</m:ci></m:apply><m:ci>m</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">33</span>)</td>
	</tr>
      
      </table></div>

    </div>
  
    <div class="section" id="S5">
    <h2 class="title section-title"><span class="refnum"><span class="reftype">§ </span>5. </span>Operator-valued deformed Stirling numbers</h2>
  <div class="para" id="S5.p1">
    <p class="p">In the present section, we attempt to derive
the normally ordered expansion of a power of the number operator for
arbitrarily deformed bosons.
Allowing <m:math display="inline"><m:semantics><m:mi>p</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>p</m:ci></m:annotation-xml></m:semantics></m:math>, <m:math display="inline"><m:semantics><m:mi>q</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>q</m:ci></m:annotation-xml></m:semantics></m:math> and <m:math display="inline"><m:semantics><m:mi>Q</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>Q</m:ci></m:annotation-xml></m:semantics></m:math> to be arbitrary,
we demand</p>
  <table class="equation" id="S5.E34">
      
      
	<tr valign="baseline" class="equation" id="S5.E34">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msubsup><m:mfenced open="[" close="]"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced><m:mi>G</m:mi><m:mi>m</m:mi></m:msubsup><m:mo>=</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>k</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>m</m:mi></m:mover><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:mover accent="true"><m:mi>S</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:ci>⌃</m:ci><m:ci>S</m:ci></m:apply><m:apply><m:vector/><m:ci>m</m:ci><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">34</span>)</td>
	</tr>
      
      </table>
    <p class="p">Using the general relation (<a href="#S3.E24" title="Eq.24 in §3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">24</a>),
we derive the recurrence relation</p>
  <table class="equation" id="S5.E35">
      
      
	<tr valign="baseline" class="equation" id="S5.E35">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:mover accent="true"><m:mi>S</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:mi>m</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo>,</m:mo><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mrow><m:msup><m:mi>Q</m:mi><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:mover accent="true"><m:mi>S</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow><m:mo>,</m:mo><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:mrow></m:mfenced></m:mrow><m:mo>+</m:mo><m:mrow><m:mover accent="true"><m:mi>S</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mi mathvariant="normal">Φ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:ci>⌃</m:ci><m:ci>S</m:ci></m:apply><m:apply><m:vector/><m:apply><m:plus/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply><m:apply><m:plus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>Q</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:apply><m:ci>⌃</m:ci><m:ci>S</m:ci></m:apply><m:apply><m:vector/><m:ci>m</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply><m:apply><m:plus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:cn>1</m:cn></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:ci>⌃</m:ci><m:ci>S</m:ci></m:apply><m:apply><m:vector/><m:ci>m</m:ci><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:ci>Φ</m:ci><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">35</span>)</td>
	</tr>
      
      </table>
    <p class="p">The “boundary conditions” and initial values, for all values of
<m:math display="inline"><m:semantics><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>, are the same as those following Eq. (<a href="#S2.E4" title="Eq.4 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">4</a>).</p>
  </div>

  <div class="para" id="S5.p2">
    <p class="p">The M-type bosons (<m:math display="inline"><m:semantics><m:mrow><m:mi>p</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>p</m:ci><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math>), with the choice <m:math display="inline"><m:semantics><m:mrow><m:mi>Q</m:mi><m:mo>=</m:mo><m:mi>q</m:mi></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>Q</m:ci><m:ci>q</m:ci></m:apply></m:annotation-xml></m:semantics></m:math> which yields
<m:math display="inline"><m:semantics><m:mrow><m:mrow><m:msub><m:mi mathvariant="normal">Φ</m:mi><m:mi>M</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>M</m:mi></m:msub></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>Φ</m:ci><m:ci>M</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>M</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>, were studied in section 4.
For this case, <m:math display="inline"><m:semantics><m:mrow><m:mover accent="true"><m:mi>S</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:apply><m:ci>⌃</m:ci><m:ci>S</m:ci></m:apply><m:apply><m:vector/><m:ci>m</m:ci><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> does not depend on
<m:math display="inline"><m:semantics><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>. More specifically,
Eq. (<a href="#S5.E35" title="Eq.35 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">35</a>) then reduces
to Eq. (<a href="#S2.E9" title="Eq.9 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">9</a>). For the
G-type bosons, we found in section 3 that by choosing <m:math display="inline"><m:semantics><m:mrow><m:mi>Q</m:mi><m:mo>=</m:mo><m:mi>q</m:mi></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>Q</m:ci><m:ci>q</m:ci></m:apply></m:annotation-xml></m:semantics></m:math> we
obtain <m:math display="inline"><m:semantics><m:mrow><m:mrow><m:msub><m:mi mathvariant="normal">Φ</m:mi><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:msup><m:mi>p</m:mi><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:msup></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>Φ</m:ci><m:ci>G</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>G</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> ; consequently,
we have</p>
  <table class="equation" id="S5.E36">
      
      
	<tr valign="baseline" class="equation" id="S5.E36">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:msub><m:mover accent="true"><m:mi>S</m:mi><m:mo>⌃</m:mo></m:mover><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:mi>m</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo>,</m:mo><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mrow><m:msup><m:mi>q</m:mi><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msub><m:mover accent="true"><m:mi>S</m:mi><m:mo>⌃</m:mo></m:mover><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow><m:mo>,</m:mo><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:mrow></m:mfenced></m:mrow><m:mo>+</m:mo><m:mrow><m:msub><m:mover accent="true"><m:mi>S</m:mi><m:mo>⌃</m:mo></m:mover><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:msup><m:mi>p</m:mi><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:msup></m:mrow></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>S</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:vector/><m:apply><m:plus/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply><m:apply><m:plus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>S</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:vector/><m:ci>m</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply><m:apply><m:plus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:cn>1</m:cn></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>S</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:vector/><m:ci>m</m:ci><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>G</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">36</span>)</td>
	</tr>
      
      </table>
    <p class="p">Note that in the general case
<m:math display="inline"><m:semantics><m:mrow><m:msub><m:mover accent="true"><m:mi>S</m:mi><m:mo>⌃</m:mo></m:mover><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>S</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:vector/><m:ci>m</m:ci><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> depends on the operator <m:math display="inline"><m:semantics><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>.
The special cases <m:math display="inline"><m:semantics><m:mrow><m:mi>p</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>p</m:ci><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math> and <m:math display="inline"><m:semantics><m:mrow><m:mi>p</m:mi><m:mo>=</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>p</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:cn>1</m:cn></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> are contained in
Eq. (<a href="#S5.E36" title="Eq.36 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">36</a>).
The dependence of <m:math display="inline"><m:semantics><m:mrow><m:msub><m:mover accent="true"><m:mi>S</m:mi><m:mo>⌃</m:mo></m:mover><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>S</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:vector/><m:ci>m</m:ci><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> on <m:math display="inline"><m:semantics><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:annotation-xml></m:semantics></m:math> for all cases
except <m:math display="inline"><m:semantics><m:mrow><m:mi>p</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>p</m:ci><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math> can be taken to imply that we have
actually failed to obtain a normally
ordered expansion for <m:math display="inline"><m:semantics><m:msubsup><m:mfenced open="[" close="]"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced><m:mi>G</m:mi><m:mi>m</m:mi></m:msubsup><m:annotation-xml encoding="MathML-Content"><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:ci>m</m:ci></m:apply></m:annotation-xml></m:semantics></m:math> in terms of a finite sum in
<m:math display="inline"><m:semantics><m:mrow><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> with <m:math display="inline"><m:semantics><m:mrow><m:mi>k</m:mi><m:mo>=</m:mo><m:mrow><m:mn>1</m:mn><m:mo>,</m:mo><m:mn> 2</m:mn><m:mo>,</m:mo><m:mi mathvariant="normal">⋯</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>k</m:ci><m:apply><m:list/><m:cn>1</m:cn><m:cn> 2</m:cn><m:ci>⋯</m:ci><m:ci>m</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>.</p>
  </div>

  <div class="para" id="S5.p3">
    <p class="p">The structure of the recurrence  relation (<a href="#S5.E36" title="Eq.36 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">36</a>) indicates
that the dependence on <m:math display="inline"><m:semantics><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:annotation-xml></m:semantics></m:math> of the deformed Stirling
numbers <m:math display="inline"><m:semantics><m:mrow><m:msub><m:mover accent="true"><m:mi>S</m:mi><m:mo>⌃</m:mo></m:mover><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>S</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:vector/><m:ci>m</m:ci><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>
can be expressed in terms of the factor
<m:math display="inline"><m:semantics><m:msup><m:mi>p</m:mi><m:mrow><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>-</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:msup><m:annotation-xml encoding="MathML-Content"><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:times/><m:apply><m:minus/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>. Defining the (<m:math display="inline"><m:semantics><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>-independent) reduced
Stirling numbers of the second kind
<m:math display="inline"><m:semantics><m:mrow><m:mi mathvariant="normal">Ξ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:ci>Ξ</m:ci><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> through</p>
  <table class="equation" id="S5.E37">
      
      
	<tr valign="baseline" class="equation" id="S5.E37">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msub><m:mover accent="true"><m:mi>S</m:mi><m:mo>⌃</m:mo></m:mover><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msup><m:mi>q</m:mi><m:mrow><m:mrow><m:mi>k</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>p</m:mi><m:mrow><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>-</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:msup><m:mo>⁢</m:mo><m:mi mathvariant="normal">Ξ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>S</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:vector/><m:ci>m</m:ci><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:divide/><m:apply><m:times/><m:ci>k</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:times/><m:apply><m:minus/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply><m:ci>Ξ</m:ci><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">37</span>)</td>
	</tr>
      
      </table>
    <p class="p">we obtain the recurrence relation</p>
  <table class="equation" id="S5.E38">
      
      
	<tr valign="baseline" class="equation" id="S5.E38">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mi mathvariant="normal">Ξ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:mi>m</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mrow><m:msup><m:mi>p</m:mi><m:mrow><m:mi>m</m:mi><m:mo>-</m:mo><m:mi>k</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:mi mathvariant="normal">Ξ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mrow></m:mfenced></m:mrow><m:mo>+</m:mo><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mi mathvariant="normal">Ξ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:ci>Ξ</m:ci><m:apply><m:interval closure="open"/><m:apply><m:plus/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply><m:ci>k</m:ci></m:apply></m:apply><m:apply><m:plus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:plus/><m:apply><m:minus/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply><m:cn>1</m:cn></m:apply></m:apply><m:ci>Ξ</m:ci><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>G</m:ci></m:apply><m:ci>Ξ</m:ci><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">38</span>)</td>
	</tr>
      
      </table>
    <p class="p">with  the initial condition <m:math display="inline"><m:semantics><m:mrow><m:mrow><m:mi mathvariant="normal">Ξ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mn>1</m:mn><m:mo>,</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:ci>Ξ</m:ci><m:apply><m:interval closure="open"/><m:cn>1</m:cn><m:cn>1</m:cn></m:apply></m:apply><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math>.</p>
  </div>

  <div class="para" id="S5.p4">
    <p class="p">To obtain
the “inverse relation” to (<a href="#S5.E34" title="Eq.34 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">34</a>),
expressing a normally ordered term
<m:math display="inline"><m:semantics><m:mrow><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> by means of a polynomial in <m:math display="inline"><m:semantics><m:msub><m:mfenced open="[" close="]"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced><m:mi>G</m:mi></m:msub><m:annotation-xml encoding="MathML-Content"><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>G</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>,
we need the “G-arithmetic” identity</p>
  <table class="equation" id="S5.E39">
      
      
	<tr valign="baseline" class="equation" id="S5.E39">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>a</m:mi><m:mo>-</m:mo><m:mi>b</m:mi></m:mrow></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>=</m:mo><m:mrow><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mi>b</m:mi></m:mrow></m:msup><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>a</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>-</m:mo><m:mrow><m:msup><m:mi>p</m:mi><m:mrow><m:mi>a</m:mi><m:mo>-</m:mo><m:mi>b</m:mi></m:mrow></m:msup><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>b</m:mi></m:mfenced><m:mi>G</m:mi></m:msub></m:mrow></m:mrow></m:mfenced></m:mrow></m:mrow><m:mo>,</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:minus/><m:ci>a</m:ci><m:ci>b</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:ci>b</m:ci></m:apply></m:apply><m:apply><m:minus/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>a</m:ci><m:ci>G</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:minus/><m:ci>a</m:ci><m:ci>b</m:ci></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>b</m:ci><m:ci>G</m:ci></m:apply></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">39</span>)</td>
	</tr>
      
      </table>
    <p class="p">which follows from the two identities</p>
  <table class="equation" id="S5.E40">
      
      
	<tr valign="baseline" class="equation" id="S5.E40">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>a</m:mi><m:mo>+</m:mo><m:mi>b</m:mi></m:mrow></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>=</m:mo><m:mrow><m:mrow><m:msup><m:mi>q</m:mi><m:mi>b</m:mi></m:msup><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>a</m:mi></m:mfenced><m:mi>G</m:mi></m:msub></m:mrow><m:mo>+</m:mo><m:mrow><m:msup><m:mi>p</m:mi><m:mi>a</m:mi></m:msup><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>b</m:mi></m:mfenced><m:mi>G</m:mi></m:msub></m:mrow></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:plus/><m:ci>a</m:ci><m:ci>b</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:plus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:ci>b</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>a</m:ci><m:ci>G</m:ci></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:ci>a</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>b</m:ci><m:ci>G</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">40</span>)</td>
	</tr>
      
      </table>
    <p class="p">and</p>
  <table class="equation" id="S5.E41">
      
      
	<tr valign="baseline" class="equation" id="S5.E41">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mo>-</m:mo><m:mi>b</m:mi></m:mrow></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>=</m:mo><m:mrow><m:mo>-</m:mo><m:mrow><m:msup><m:mfenced open="(" close=")"><m:mrow><m:mi>p</m:mi><m:mo>⁢</m:mo><m:mi>q</m:mi></m:mrow></m:mfenced><m:mrow><m:mo>-</m:mo><m:mi>b</m:mi></m:mrow></m:msup><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>b</m:mi></m:mfenced><m:mi>G</m:mi></m:msub></m:mrow></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:minus/><m:ci>b</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:minus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:times/><m:ci>p</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:minus/><m:ci>b</m:ci></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>b</m:ci><m:ci>G</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">41</span>)</td>
	</tr>
      
      </table>
    <p class="p">We now proceed to obtain the desired relation</p>
  <table class="equation" id="S5.E42">
      
      
	<tr valign="baseline" class="equation" id="S5.E42">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow><m:mo>=</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>m</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>k</m:mi></m:mover><m:msub><m:mover accent="true"><m:mi>s</m:mi><m:mo>⌃</m:mo></m:mover><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:msubsup><m:mfenced open="[" close="]"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced><m:mi>G</m:mi><m:mi>m</m:mi></m:msubsup></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>s</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:vector/><m:ci>k</m:ci><m:ci>m</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:ci>m</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">42</span>)</td>
	</tr>
      
      </table>
    <p class="p">Since  <m:math display="inline"><m:semantics><m:mrow><m:mrow><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mrow><m:mi>k</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mrow><m:mi>k</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:msup></m:mrow><m:mo>=</m:mo><m:mrow><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow><m:mo>=</m:mo><m:mrow><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>-</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced><m:mi>G</m:mi></m:msub></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:apply><m:plus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:apply><m:plus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply></m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:minus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:ci>G</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>,
we can use Eqs. (<a href="#S5.E39" title="Eq.39 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">39</a>) and (<a href="#S5.E42" title="Eq.42 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">42</a>) to obtain
the recurrence relation</p>
  <table class="equation" id="S5.E43">
      
      
	<tr valign="baseline" class="equation" id="S5.E43">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:msub><m:mover accent="true"><m:mi>s</m:mi><m:mo>⌃</m:mo></m:mover><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:mi>k</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo>,</m:mo><m:mi>m</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mi>k</m:mi></m:mrow></m:msup><m:mo>⁢</m:mo><m:none/><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:msub><m:mover accent="true"><m:mi>s</m:mi><m:mo>⌃</m:mo></m:mover><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mrow><m:mi>m</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:mo>-</m:mo><m:mrow><m:msup><m:mi>p</m:mi><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>-</m:mo><m:mi>k</m:mi></m:mrow></m:msup><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:msub><m:mover accent="true"><m:mi>s</m:mi><m:mo>⌃</m:mo></m:mover><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:none/></m:mrow></m:mrow></m:mfenced></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>s</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:vector/><m:apply><m:plus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply><m:ci>m</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:ci>k</m:ci></m:apply></m:apply><m:ci/><m:apply><m:minus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>s</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:vector/><m:ci>k</m:ci><m:apply><m:minus/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:minus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>k</m:ci></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>G</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>s</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:vector/><m:ci>k</m:ci><m:ci>m</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:ci/></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">43</span>)</td>
	</tr>
      
      </table>
    <p class="p">Note that for <m:math display="inline"><m:semantics><m:mrow><m:mi>p</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>p</m:ci><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math> this recurrence relation reduces to
Eq. (<a href="#S2.E8" title="Eq.8 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">8</a>).</p>
  </div>

  <div class="para" id="S5.p5">
    <p class="p">Introducing the (<m:math display="inline"><m:semantics><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>-independent) reduced Stirling numbers
of the first kind
<m:math display="inline"><m:semantics><m:mrow><m:mi>ξ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:ci>ξ</m:ci><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> such that</p>
  <table class="equation" id="S5.E44">
      
      
	<tr valign="baseline" class="equation" id="S5.E44">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msub><m:mover accent="true"><m:mi>s</m:mi><m:mo>⌃</m:mo></m:mover><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mrow><m:mrow><m:mi>k</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:mrow></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>p</m:mi><m:mrow><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:msup><m:mo>⁢</m:mo><m:mi>ξ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>s</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:vector/><m:ci>k</m:ci><m:ci>m</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:apply><m:divide/><m:apply><m:times/><m:ci>k</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:times/><m:apply><m:minus/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply><m:ci>ξ</m:ci><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">44</span>)</td>
	</tr>
      
      </table>
    <p class="p">in Eq. (<a href="#S5.E43" title="Eq.43 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">43</a>), we obtain the recurrence relation</p>
  <table class="equation" id="S5.E45">
      
      
	<tr valign="baseline" class="equation" id="S5.E45">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:mi>ξ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:mi>k</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mrow><m:mi>ξ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mrow><m:mi>m</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mrow></m:mfenced></m:mrow><m:mo>-</m:mo><m:mrow><m:msup><m:mi>p</m:mi><m:mrow><m:mo>-</m:mo><m:mi>k</m:mi></m:mrow></m:msup><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mi>ξ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced></m:mrow></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:ci>ξ</m:ci><m:apply><m:interval closure="open"/><m:apply><m:plus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply><m:ci>m</m:ci></m:apply></m:apply><m:apply><m:minus/><m:apply><m:times/><m:ci>ξ</m:ci><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:apply><m:minus/><m:ci>m</m:ci><m:cn>1</m:cn></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:minus/><m:ci>k</m:ci></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>G</m:ci></m:apply><m:ci>ξ</m:ci><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">45</span>)</td>
	</tr>
      
      </table></div>

  <div class="para" id="S5.p6">
    <p class="p">The exponential dependence on <m:math display="inline"><m:semantics><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:annotation-xml></m:semantics></m:math> of the
deformed Stirling numbers of the first kind, <m:math display="inline"><m:semantics><m:mrow><m:msub><m:mover accent="true"><m:mi>s</m:mi><m:mo>⌃</m:mo></m:mover><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>s</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:vector/><m:ci>k</m:ci><m:ci>m</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>,
means that we have not been able to
express <m:math display="inline"><m:semantics><m:mrow><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> as
a <span style="" class="text italic">polynomial</span> in <m:math display="inline"><m:semantics><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:annotation-xml></m:semantics></m:math> but
we did express it as a
<span style="" class="text italic">function</span> of <m:math display="inline"><m:semantics><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>.</p>
  </div>

  <div class="para" id="S5.p7">
    <p class="p">In order to derive the bi-orthogonality relations between the
deformed Stirling numbers of the first and second kinds, we first
rewrite Eq. (<a href="#S5.E34" title="Eq.34 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">34</a>) in the form</p>
  <table class="equation" id="S5.E46">
      
      
	<tr valign="baseline" class="equation" id="S5.E46">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msubsup><m:mfenced open="[" close="]"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced><m:mi>G</m:mi><m:mi>m</m:mi></m:msubsup><m:mo>=</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>k</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>m</m:mi></m:mover><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msub><m:mover accent="true"><m:mi>S</m:mi><m:mo>⌃</m:mo></m:mover><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi><m:mo>,</m:mo><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>-</m:mo><m:mi>k</m:mi></m:mrow></m:mrow></m:mfenced></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>S</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:vector/><m:ci>m</m:ci><m:ci>k</m:ci><m:apply><m:minus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">46</span>)</td>
	</tr>
      
      </table>
    <p class="p">Using Eq. (<a href="#S5.E37" title="Eq.37 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">37</a>) we obtain</p>
  <table class="equation" id="S5.E47">
      
      
	<tr valign="baseline" class="equation" id="S5.E47">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:msub><m:mover accent="true"><m:mi>S</m:mi><m:mo>⌃</m:mo></m:mover><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi><m:mo>,</m:mo><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>-</m:mo><m:mi>k</m:mi></m:mrow></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msup><m:mi>p</m:mi><m:mrow><m:mi>k</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced></m:mrow></m:msup><m:mo>⁢</m:mo><m:msub><m:mover accent="true"><m:mi>S</m:mi><m:mo>⌃</m:mo></m:mover><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi><m:mo>,</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>S</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:vector/><m:ci>m</m:ci><m:ci>k</m:ci><m:apply><m:minus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>k</m:ci></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:times/><m:ci>k</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:ci>⌃</m:ci><m:ci>S</m:ci></m:apply><m:ci>G</m:ci></m:apply><m:apply><m:vector/><m:ci>m</m:ci><m:ci>k</m:ci><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">47</span>)</td>
	</tr>
      
      </table>
    <p class="p">Defining <m:math display="inline"><m:semantics><m:mrow><m:mrow><m:msup><m:mi mathvariant="normal">Ξ</m:mi><m:mo>′</m:mo></m:msup><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msup><m:mi>p</m:mi><m:mrow><m:mi>k</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced></m:mrow></m:msup><m:mo>⁢</m:mo><m:mi mathvariant="normal">Ξ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>Ξ</m:ci><m:ci>′</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:times/><m:ci>k</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply></m:apply></m:apply><m:ci>Ξ</m:ci><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>,
we obtain relations of the form of Eqs. (<a href="#S2.E16" title="Eq.16 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">16</a>) and (<a href="#S2.E17" title="Eq.17 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">17</a>)
with <m:math display="inline"><m:semantics><m:mrow><m:msup><m:mi mathvariant="normal">Ξ</m:mi><m:mo>′</m:mo></m:msup><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>Ξ</m:ci><m:ci>′</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> replacing <m:math display="inline"><m:semantics><m:mrow><m:msub><m:mi>S</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>m</m:mi><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>S</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>m</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>
and <m:math display="inline"><m:semantics><m:mrow><m:mi>ξ</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:ci>ξ</m:ci><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> replacing <m:math display="inline"><m:semantics><m:mrow><m:msub><m:mi>s</m:mi><m:mi>q</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>,</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>s</m:ci><m:ci>q</m:ci></m:apply><m:apply><m:interval closure="open"/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>.
</p>
  </div>

    </div>
  
    <div class="section" id="S6">
    <h2 class="title section-title"><span class="refnum"><span class="reftype">§ </span>6. </span>A generating function for the deformed Stirling numbers
of the first kind</h2>
  <div class="para" id="S6.p1">
    <p class="p">We start by transforming the <m:math display="inline"><m:semantics><m:mi>q</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>q</m:ci></m:annotation-xml></m:semantics></m:math>-binomial theorem <cite class="cite">[<a href="#bib.bib13" title="" class="ref">13</a>]</cite> into a
G-binomial theorem. By introducing the symbol</p>
  <table class="equation" id="S6.E48">
      
      
	<tr valign="baseline" class="equation" id="S6.E48">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:msup><m:mfenced open="(" close=")"><m:mrow><m:mi>λ</m:mi><m:mo>;</m:mo><m:mi>x</m:mi></m:mrow></m:mfenced><m:mfenced open="(" close=")"><m:mi>l</m:mi></m:mfenced></m:msup><m:mo>=</m:mo><m:mrow><m:mfenced open="(" close=")"><m:mrow><m:mi>λ</m:mi><m:mo>+</m:mo><m:mi>x</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:mi>p</m:mi><m:mo>⁢</m:mo><m:mi>λ</m:mi></m:mrow><m:mo>+</m:mo><m:mrow><m:mi>q</m:mi><m:mo>⁢</m:mo><m:mi>x</m:mi></m:mrow></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:msup><m:mi>p</m:mi><m:mn>2</m:mn></m:msup><m:mo>⁢</m:mo><m:mi>λ</m:mi></m:mrow><m:mo>+</m:mo><m:mrow><m:msup><m:mi>q</m:mi><m:mn>2</m:mn></m:msup><m:mo>⁢</m:mo><m:mi>x</m:mi></m:mrow></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mi mathvariant="normal">⋯</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:msup><m:mi>p</m:mi><m:mrow><m:mi>l</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:mi>λ</m:mi></m:mrow><m:mo>+</m:mo><m:mrow><m:msup><m:mi>q</m:mi><m:mrow><m:mi>l</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:mi>x</m:mi></m:mrow></m:mrow></m:mfenced></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:interval closure="open"/><m:ci>λ</m:ci><m:ci>x</m:ci></m:apply><m:ci>l</m:ci></m:apply><m:apply><m:times/><m:apply><m:plus/><m:ci>λ</m:ci><m:ci>x</m:ci></m:apply><m:apply><m:plus/><m:apply><m:times/><m:ci>p</m:ci><m:ci>λ</m:ci></m:apply><m:apply><m:times/><m:ci>q</m:ci><m:ci>x</m:ci></m:apply></m:apply><m:apply><m:plus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:cn>2</m:cn></m:apply><m:ci>λ</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:cn>2</m:cn></m:apply><m:ci>x</m:ci></m:apply></m:apply><m:ci>⋯</m:ci><m:apply><m:plus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:minus/><m:ci>l</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>λ</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:ci>l</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>x</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">48</span>)</td>
	</tr>
      
      </table>
    <p class="p">we have</p>
  <table class="equation" id="S6.E49">
      
      
	<tr valign="baseline" class="equation" id="S6.E49">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msup><m:mfenced open="(" close=")"><m:mrow><m:mi>λ</m:mi><m:mo>;</m:mo><m:mi>x</m:mi></m:mrow></m:mfenced><m:mfenced open="(" close=")"><m:mi>l</m:mi></m:mfenced></m:msup><m:mo>=</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>i</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>0</m:mn></m:mrow></m:munder><m:mi>l</m:mi></m:mover><m:msub><m:mfenced open="[" close="]"><m:mtable rowspacing="0.2ex" columnspacing="0.4em"><m:mtr><m:mtd/><m:mtd columnalign="center"><m:mi>l</m:mi></m:mtd><m:mtd/></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="center"><m:mi>i</m:mi></m:mtd><m:mtd/></m:mtr></m:mtable></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:msup><m:mi>p</m:mi><m:mrow><m:mrow><m:mi>i</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>i</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mrow><m:mfenced open="(" close=")"><m:mrow><m:mi>l</m:mi><m:mo>-</m:mo><m:mi>i</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>l</m:mi><m:mo>-</m:mo><m:mi>i</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>λ</m:mi><m:mi>i</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>x</m:mi><m:mrow><m:mi>l</m:mi><m:mo>-</m:mo><m:mi>i</m:mi></m:mrow></m:msup></m:mrow></m:mrow><m:mo>,</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:interval closure="open"/><m:ci>λ</m:ci><m:ci>x</m:ci></m:apply><m:ci>l</m:ci></m:apply><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>i</m:ci><m:cn>0</m:cn></m:apply></m:apply><m:ci>l</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:mtext>li</m:mtext><m:ci>G</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:divide/><m:apply><m:times/><m:ci>i</m:ci><m:apply><m:minus/><m:ci>i</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:divide/><m:apply><m:times/><m:apply><m:minus/><m:ci>l</m:ci><m:ci>i</m:ci></m:apply><m:apply><m:minus/><m:ci>l</m:ci><m:ci>i</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>λ</m:ci><m:ci>i</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>x</m:ci><m:apply><m:minus/><m:ci>l</m:ci><m:ci>i</m:ci></m:apply></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">49</span>)</td>
	</tr>
      
      </table>
    <p class="p">where</p>
  <table class="equation" id="S6.E50">
      
      
	<tr valign="baseline" class="equation" id="S6.E50">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mtable rowspacing="0.2ex" columnspacing="0.4em"><m:mtr><m:mtd/><m:mtd columnalign="center"><m:mi>l</m:mi></m:mtd><m:mtd/></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="center"><m:mi>i</m:mi></m:mtd><m:mtd/></m:mtr></m:mtable></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>=</m:mo><m:mfrac><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>l</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mi mathvariant="normal">!</m:mi></m:mrow><m:mrow><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>i</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mi mathvariant="normal">!</m:mi></m:mrow><m:mo>⁢</m:mo><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>l</m:mi><m:mo>-</m:mo><m:mi>i</m:mi></m:mrow></m:mfenced><m:mi>G</m:mi></m:msub><m:mi mathvariant="normal">!</m:mi></m:mrow></m:mrow></m:mfrac></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:mtext>li</m:mtext><m:ci>G</m:ci></m:apply><m:apply><m:divide/><m:apply><m:factorial/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>l</m:ci><m:ci>G</m:ci></m:apply></m:apply><m:apply><m:times/><m:apply><m:factorial/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>i</m:ci><m:ci>G</m:ci></m:apply></m:apply><m:apply><m:factorial/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:minus/><m:ci>l</m:ci><m:ci>i</m:ci></m:apply><m:ci>G</m:ci></m:apply></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">50</span>)</td>
	</tr>
      
      </table>
    <p class="p">is a G-binomial coefficient and
<m:math display="inline"><m:semantics><m:mrow><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mi mathvariant="normal">!</m:mi></m:mrow><m:mo>=</m:mo><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mn>1</m:mn></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mn>2</m:mn></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mi mathvariant="normal">⋯</m:mi><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>G</m:mi></m:msub></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:factorial/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>G</m:ci></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:cn>1</m:cn><m:ci>G</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:cn>2</m:cn><m:ci>G</m:ci></m:apply><m:ci>⋯</m:ci><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>G</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>.
Equation (<a href="#S6.E49" title="Eq.49 in §6. A generating function for the deformed Stirling numbers&#10;of the first kind in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">49</a>) can be proved by induction, using the
G-binomial coefficient recurrence relation</p>
  <table class="equation" id="S6.E51">
      
      
	<tr valign="baseline" class="equation" id="S6.E51">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mtable rowspacing="0.2ex" columnspacing="0.4em"><m:mtr><m:mtd columnalign="center"><m:mi>l</m:mi></m:mtd><m:mtd columnalign="center"><m:mo>+</m:mo></m:mtd><m:mtd columnalign="center"><m:mn>1</m:mn></m:mtd></m:mtr><m:mtr><m:mtd columnalign="center"><m:none/></m:mtd><m:mtd columnalign="center"><m:mi>i</m:mi></m:mtd><m:mtd columnalign="center"><m:none/></m:mtd></m:mtr></m:mtable></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>=</m:mo><m:mrow><m:mrow><m:msup><m:mi>p</m:mi><m:mrow><m:mi>l</m:mi><m:mo>+</m:mo><m:mn>1</m:mn><m:mo>-</m:mo><m:mi>i</m:mi></m:mrow></m:msup><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mtable rowspacing="0.2ex" columnspacing="0.4em"><m:mtr><m:mtd columnalign="center"><m:none/></m:mtd><m:mtd columnalign="center"><m:mi>l</m:mi></m:mtd><m:mtd columnalign="center"><m:none/></m:mtd></m:mtr><m:mtr><m:mtd columnalign="center"><m:mi>i</m:mi></m:mtd><m:mtd columnalign="center"><m:mo>-</m:mo></m:mtd><m:mtd columnalign="center"><m:mn>1</m:mn></m:mtd></m:mtr></m:mtable></m:mfenced><m:mi>G</m:mi></m:msub></m:mrow><m:mo>+</m:mo><m:mrow><m:msup><m:mi>q</m:mi><m:mi>i</m:mi></m:msup><m:mo>⁢</m:mo><m:msub><m:mfenced open="[" close="]"><m:mtable rowspacing="0.2ex" columnspacing="0.4em"><m:mtr><m:mtd/><m:mtd columnalign="center"><m:mi>l</m:mi></m:mtd><m:mtd/></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="center"><m:mi>i</m:mi></m:mtd><m:mtd/></m:mtr></m:mtable></m:mfenced><m:mi>G</m:mi></m:msub></m:mrow></m:mrow></m:mrow><m:mo>,</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:mtext>l+1i</m:mtext><m:ci>G</m:ci></m:apply><m:apply><m:plus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:minus/><m:apply><m:plus/><m:ci>l</m:ci><m:cn>1</m:cn></m:apply><m:ci>i</m:ci></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:mtext>li-1</m:mtext><m:ci>G</m:ci></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:ci>i</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:mtext>li</m:mtext><m:ci>G</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">51</span>)</td>
	</tr>
      
      </table>
    <p class="p">which follows from the definition of the G-binomial coefficient
on using the G-arithmetic relation (<a href="#S5.E40" title="Eq.40 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">40</a>).</p>
  </div>

  <div class="para" id="S6.p2">
    <p class="p">Now, from the identity</p>
  <table class="equation" id="S6.E52">
      
      
	<tr valign="baseline" class="equation" id="S6.E52">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mfrac><m:mrow><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mi mathvariant="normal">!</m:mi></m:mrow></m:mfrac><m:mo>⁢</m:mo><m:mfenced open="|" close="&gt;"><m:mi>l</m:mi></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mtable rowspacing="0.2ex" columnspacing="0.4em"><m:mtr><m:mtd/><m:mtd columnalign="center"><m:mi>l</m:mi></m:mtd><m:mtd/></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="center"><m:mi>k</m:mi></m:mtd><m:mtd/></m:mtr></m:mtable></m:mfenced><m:mi>G</m:mi></m:msub><m:mo>⁢</m:mo><m:mfenced open="|" close="&gt;"><m:mi>l</m:mi></m:mfenced></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:divide/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply><m:apply><m:factorial/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>G</m:ci></m:apply></m:apply></m:apply><m:apply><m:ci/><m:ci>l</m:ci></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:mtext>lk</m:mtext><m:ci>G</m:ci></m:apply><m:apply><m:ci/><m:ci>l</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">52</span>)</td>
	</tr>
      
      </table>
    <p class="p">we obtain</p>
  <table class="equation" id="S6.E53">
      
      
	<tr valign="baseline" class="equation" id="S6.E53">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>k</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>0</m:mn></m:mrow></m:munder><m:mi>m</m:mi></m:mover><m:msup><m:mi>p</m:mi><m:mrow><m:mrow><m:mi>k</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mrow><m:mfenced open="(" close=")"><m:mrow><m:mi>l</m:mi><m:mo>-</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>l</m:mi><m:mo>-</m:mo><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>λ</m:mi><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:mfrac><m:mrow><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mi mathvariant="normal">!</m:mi></m:mrow></m:mfrac><m:mo>⁢</m:mo><m:mfenced open="|" close="&gt;"><m:mi>l</m:mi></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:msup><m:mfenced open="(" close=")"><m:mrow><m:mi>λ</m:mi><m:mo>;</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced><m:mfenced open="(" close=")"><m:mi>l</m:mi></m:mfenced></m:msup><m:mo>⁢</m:mo><m:mfenced open="|" close="&gt;"><m:mi>l</m:mi></m:mfenced></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>k</m:ci><m:cn>0</m:cn></m:apply></m:apply><m:ci>m</m:ci></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:divide/><m:apply><m:times/><m:ci>k</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:divide/><m:apply><m:times/><m:apply><m:minus/><m:ci>l</m:ci><m:ci>k</m:ci></m:apply><m:apply><m:minus/><m:ci>l</m:ci><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>λ</m:ci><m:ci>k</m:ci></m:apply><m:apply><m:divide/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply><m:apply><m:factorial/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>G</m:ci></m:apply></m:apply></m:apply><m:apply><m:ci/><m:ci>l</m:ci></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:interval closure="open"/><m:ci>λ</m:ci><m:cn>1</m:cn></m:apply><m:ci>l</m:ci></m:apply><m:apply><m:ci/><m:ci>l</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">53</span>)</td>
	</tr>
      
      </table>
    <p class="p">which can be written as an operator identity</p>
  <table class="equation" id="S6.E54">
      
      
	<tr valign="baseline" class="equation" id="S6.E54">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>k</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>0</m:mn></m:mrow></m:munder><m:mi mathvariant="normal">∞</m:mi></m:mover><m:msup><m:mi>p</m:mi><m:mrow><m:mrow><m:mi>k</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mrow><m:mfenced open="(" close=")"><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>-</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>-</m:mo><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>λ</m:mi><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:mfrac><m:mrow><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mi mathvariant="normal">!</m:mi></m:mrow></m:mfrac></m:mrow><m:mo>=</m:mo><m:msup><m:mfenced open="(" close=")"><m:mrow><m:mi>λ</m:mi><m:mo>;</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced><m:mfenced open="(" close=")"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced></m:msup></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>k</m:ci><m:cn>0</m:cn></m:apply></m:apply><m:infinity/></m:apply><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:divide/><m:apply><m:times/><m:ci>k</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:divide/><m:apply><m:times/><m:apply><m:minus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:minus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>λ</m:ci><m:ci>k</m:ci></m:apply><m:apply><m:divide/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply><m:apply><m:factorial/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>G</m:ci></m:apply></m:apply></m:apply></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:interval closure="open"/><m:ci>λ</m:ci><m:cn>1</m:cn></m:apply><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">54</span>)</td>
	</tr>
      
      </table>
    <p class="p">To obtain an expression for <m:math display="inline"><m:semantics><m:mrow><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> as a function of
the number operator <m:math display="inline"><m:semantics><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>, we have to expand the right-hand side
of Eq. (<a href="#S6.E54" title="Eq.54 in §6. A generating function for the deformed Stirling numbers&#10;of the first kind in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">54</a>) in powers of <m:math display="inline"><m:semantics><m:mi>λ</m:mi><m:annotation-xml encoding="MathML-Content"><m:ci>λ</m:ci></m:annotation-xml></m:semantics></m:math>.
The coefficient of <m:math display="inline"><m:semantics><m:msup><m:mi>λ</m:mi><m:mi>k</m:mi></m:msup><m:annotation-xml encoding="MathML-Content"><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>λ</m:ci><m:ci>k</m:ci></m:apply></m:annotation-xml></m:semantics></m:math> can be extracted by writing
</p>
  <table class="equation" id="S6.E55">
      
      
	<tr valign="baseline" class="equation" id="S6.E55">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mrow><m:msup><m:mfenced open="(" close=")"><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mfenced><m:mi>k</m:mi></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mi>k</m:mi></m:msup></m:mrow><m:mo>=</m:mo><m:msub><m:mfenced open="" close="|"><m:mrow><m:mfrac><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mi>G</m:mi></m:msub><m:mi mathvariant="normal">!</m:mi></m:mrow><m:mrow><m:mi>k</m:mi><m:mi mathvariant="normal">!</m:mi></m:mrow></m:mfrac><m:mo>⁢</m:mo><m:msup><m:mi>p</m:mi><m:mrow><m:mo>-</m:mo><m:mrow><m:mrow><m:mi>k</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:mrow></m:msup><m:mo>⁢</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mrow><m:mrow><m:mfenced open="(" close=")"><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>-</m:mo><m:mi>k</m:mi></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>-</m:mo><m:mi>k</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:mrow></m:msup><m:mo>⁢</m:mo><m:mfrac><m:msup><m:mo>∂</m:mo><m:mi>k</m:mi></m:msup><m:mrow><m:mo>∂</m:mo><m:mo>⁡</m:mo><m:msup><m:mi>λ</m:mi><m:mi>k</m:mi></m:msup></m:mrow></m:mfrac><m:mo>⁢</m:mo><m:msup><m:mfenced open="(" close=")"><m:mrow><m:mi>λ</m:mi><m:mo>;</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced><m:mfenced open="(" close=")"><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mfenced></m:msup><m:mo>⁢</m:mo><m:none/></m:mrow></m:mfenced><m:mrow><m:mi>λ</m:mi><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow></m:msub></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>k</m:ci></m:apply></m:apply><m:apply><m:ci/><m:apply><m:times/><m:apply><m:divide/><m:apply><m:factorial/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:ci>G</m:ci></m:apply></m:apply><m:apply><m:factorial/><m:ci>k</m:ci></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:minus/><m:apply><m:divide/><m:apply><m:times/><m:ci>k</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:apply><m:divide/><m:apply><m:times/><m:apply><m:minus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:minus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:ci>k</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply></m:apply></m:apply><m:apply><m:divide/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:partialdiff/><m:ci>k</m:ci></m:apply><m:apply><m:partialdiff/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>λ</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:interval closure="open"/><m:ci>λ</m:ci><m:cn>1</m:cn></m:apply><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:ci/></m:apply><m:apply><m:eq/><m:ci>λ</m:ci><m:cn>0</m:cn></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">55</span>)</td>
	</tr>
      
      </table>
    <p class="p">The identities</p>
  <table class="equation" id="S6.E56">
      
      
	<tr valign="baseline" class="equation" id="S6.E56">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mi>m</m:mi></m:mfenced><m:mrow><m:mi>G</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:msup><m:mi>p</m:mi><m:mi>k</m:mi></m:msup><m:mo>,</m:mo><m:msup><m:mi>q</m:mi><m:mi>k</m:mi></m:msup></m:mrow></m:mfenced></m:mrow></m:msub><m:mo>=</m:mo><m:mfrac><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>k</m:mi><m:mo>⁢</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced><m:mrow><m:mi>G</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>p</m:mi><m:mo>,</m:mo><m:mi>q</m:mi></m:mrow></m:mfenced></m:mrow></m:msub><m:msub><m:mfenced open="[" close="]"><m:mi>k</m:mi></m:mfenced><m:mrow><m:mi>G</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>p</m:mi><m:mo>,</m:mo><m:mi>q</m:mi></m:mrow></m:mfenced></m:mrow></m:msub></m:mfrac></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>m</m:ci><m:apply><m:times/><m:ci>G</m:ci><m:apply><m:interval closure="open"/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:ci>k</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:ci>k</m:ci></m:apply></m:apply></m:apply></m:apply><m:apply><m:divide/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:times/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply><m:apply><m:times/><m:ci>G</m:ci><m:apply><m:interval closure="open"/><m:ci>p</m:ci><m:ci>q</m:ci></m:apply></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>k</m:ci><m:apply><m:times/><m:ci>G</m:ci><m:apply><m:interval closure="open"/><m:ci>p</m:ci><m:ci>q</m:ci></m:apply></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">56</span>)</td>
	</tr>
      
      </table>
    <p class="p">and</p>
  <table class="equation" id="S6.E57">
      
      
	<tr valign="baseline" class="equation" id="S6.E57">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>k</m:mi><m:mo>⁢</m:mo><m:mi>m</m:mi></m:mrow></m:mfenced><m:mrow><m:mi>G</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>p</m:mi><m:mo>,</m:mo><m:mi>q</m:mi></m:mrow></m:mfenced></m:mrow></m:msub><m:mo>=</m:mo><m:mrow><m:mover><m:munder><m:mo movablelimits="false">∑</m:mo><m:mrow><m:mi>i</m:mi><m:mo movablelimits="false">=</m:mo><m:mn>1</m:mn></m:mrow></m:munder><m:mi>k</m:mi></m:mover><m:mfenced open="(" close=")"><m:mtable rowspacing="0.2ex" columnspacing="0.4em"><m:mtr><m:mtd/><m:mtd columnalign="center"><m:mi>k</m:mi></m:mtd><m:mtd/></m:mtr><m:mtr><m:mtd/><m:mtd columnalign="center"><m:mi>i</m:mi></m:mtd><m:mtd/></m:mtr></m:mtable></m:mfenced><m:mo>⁢</m:mo><m:msup><m:mfenced open="(" close=")"><m:mrow><m:mi>q</m:mi><m:mo>-</m:mo><m:mi>p</m:mi></m:mrow></m:mfenced><m:mrow><m:mi>i</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:msubsup><m:mfenced open="[" close="]"><m:mi>m</m:mi></m:mfenced><m:mrow><m:mi>G</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>p</m:mi><m:mo>,</m:mo><m:mi>q</m:mi></m:mrow></m:mfenced></m:mrow><m:mi>i</m:mi></m:msubsup><m:mo>⁢</m:mo><m:msup><m:mi>p</m:mi><m:mrow><m:mi>m</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>k</m:mi><m:mo>-</m:mo><m:mi>i</m:mi></m:mrow></m:mfenced></m:mrow></m:msup></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:times/><m:ci>k</m:ci><m:ci>m</m:ci></m:apply><m:apply><m:times/><m:ci>G</m:ci><m:apply><m:interval closure="open"/><m:ci>p</m:ci><m:ci>q</m:ci></m:apply></m:apply></m:apply><m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:sum/><m:apply><m:eq/><m:ci>i</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:ci>k</m:ci></m:apply><m:apply><m:times/><m:mtext>ki</m:mtext><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:minus/><m:ci>q</m:ci><m:ci>p</m:ci></m:apply><m:apply><m:minus/><m:ci>i</m:ci><m:cn>1</m:cn></m:apply></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>m</m:ci><m:apply><m:times/><m:ci>G</m:ci><m:apply><m:interval closure="open"/><m:ci>p</m:ci><m:ci>q</m:ci></m:apply></m:apply></m:apply><m:ci>i</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>p</m:ci><m:apply><m:times/><m:ci>m</m:ci><m:apply><m:minus/><m:ci>k</m:ci><m:ci>i</m:ci></m:apply></m:apply></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">57</span>)</td>
	</tr>
      
      </table>
    <p class="p">are found to be useful when implementing Eq. (<a href="#S6.E55" title="Eq.55 in §6. A generating function for the deformed Stirling numbers&#10;of the first kind in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">55</a>).
(To avoid possible confusion we point out that the symbol
appearing in Eq. (<a href="#S6.E57" title="Eq.57 in §6. A generating function for the deformed Stirling numbers&#10;of the first kind in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">57</a>) is the conventional
binomial coefficient.)
Note that for the conventional bosons, for which <m:math display="inline"><m:semantics><m:mrow><m:mi>p</m:mi><m:mo>=</m:mo><m:mi>q</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:ci>p</m:ci><m:eq/><m:ci>q</m:ci><m:eq/><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math>,
Eq. (<a href="#S6.E55" title="Eq.55 in §6. A generating function for the deformed Stirling numbers&#10;of the first kind in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">55</a>) reduces to an expression <cite class="cite">[<a href="#bib.bib15" title="" class="ref">15</a>]</cite>
which can be related to the well-known generating
function for the conventional Stirling numbers of the first
kind <cite class="cite">[<a href="#bib.bib5" title="" class="ref">5</a>]</cite>.</p>
  </div>

    </div>
  
    <div class="section" id="S7">
    <h2 class="title section-title"><span class="refnum"><span class="reftype">§ </span>7. </span>Discussion</h2>
  <div class="para" id="S7.p1">
    <p class="p">In the present article we found that the normal ordering formulae
for powers of the boson number operator can be extended in a
simple and natural way to the M-type bosons, which satisfy
<m:math display="inline"><m:semantics><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>a</m:mi><m:mo>,</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mrow></m:mfenced><m:mi>q</m:mi></m:msub><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:interval closure="closed"/><m:ci>a</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply></m:apply><m:ci>q</m:ci></m:apply><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math>. However, for the P-type bosons,
which satisfy
<m:math display="inline"><m:semantics><m:mrow><m:msub><m:mfenced open="[" close="]"><m:mrow><m:mi>a</m:mi><m:mo>,</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mrow></m:mfenced><m:mi>q</m:mi></m:msub><m:mo>=</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:msup></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:apply><m:interval closure="closed"/><m:ci>a</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply></m:apply><m:ci>q</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>, as well as for the more general
G-type bosons, we found that the extension of the conventional
boson analysis results in “normal-ordering” expressions with
<m:math display="inline"><m:semantics><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:annotation-xml></m:semantics></m:math>-dependent coefficients.</p>
  </div>

  <div class="para" id="S7.p2">
    <p class="p">The marked difference between the M-type bosons and all the others
has already been noted before, in the context of the extension of the
Campbell-Baker-Hausdorff formula for products of exponential
operators <cite class="cite">[<a href="#bib.bib16" title="" class="ref">16</a>]</cite>. While the observations pointed out
above set apart the M-type bosons, the following may be taken to set
apart the P-type bosons disfavourably,
within the general set of G-type bosons:
Taking the Hamiltonian of the deformed harmonic oscillator to be
<m:math display="inline"><m:semantics><m:mrow><m:mi mathvariant="script">H</m:mi><m:mo>=</m:mo><m:mrow><m:mfrac><m:mrow><m:mi mathvariant="normal">ℏ</m:mi><m:mo>⁢</m:mo><m:msub><m:mi>ω</m:mi><m:mn>0</m:mn></m:msub></m:mrow><m:mn>2</m:mn></m:mfrac><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mrow><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup><m:mo>⁢</m:mo><m:mi>a</m:mi></m:mrow><m:mo>+</m:mo><m:mrow><m:mi>a</m:mi><m:mo>⁢</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mrow></m:mrow></m:mfenced></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>H</m:ci><m:apply><m:times/><m:apply><m:divide/><m:apply><m:times/><m:ci>ℏ</m:ci><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>ω</m:ci><m:cn>0</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply><m:apply><m:plus/><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>a</m:ci></m:apply><m:apply><m:times/><m:ci>a</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>
and expanding in
powers of <m:math display="inline"><m:semantics><m:mrow><m:mi>s</m:mi><m:mo>=</m:mo><m:mrow><m:mi>ln</m:mi><m:mo>⁡</m:mo><m:mi>q</m:mi></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>s</m:ci><m:apply><m:ln/><m:ci>q</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> and <m:math display="inline"><m:semantics><m:mrow><m:mi>t</m:mi><m:mo>=</m:mo><m:mrow><m:mi>ln</m:mi><m:mo>⁡</m:mo><m:mi>p</m:mi></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>t</m:ci><m:apply><m:ln/><m:ci>p</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> (which we assume to be sufficiently
small), we find that</p>
  <table class="equation" id="S7.E58">
      
      
	<tr valign="baseline" class="equation" id="S7.E58">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mi mathvariant="script">H</m:mi><m:mo>=</m:mo><m:mrow><m:mi mathvariant="normal">ℏ</m:mi><m:mo>⁢</m:mo><m:msub><m:mi>ω</m:mi><m:mn>0</m:mn></m:msub><m:mo>⁢</m:mo><m:mfenced open="[" close="]"><m:mrow><m:mfrac><m:mrow><m:mi>s</m:mi><m:mo>+</m:mo><m:mi>t</m:mi></m:mrow><m:mn>8</m:mn></m:mfrac><m:mo>+</m:mo><m:mrow><m:mfenced open="(" close=")"><m:mrow><m:mn>1</m:mn><m:mo>-</m:mo><m:mfrac><m:mrow><m:mi>s</m:mi><m:mo>+</m:mo><m:mi>t</m:mi></m:mrow><m:mn>2</m:mn></m:mfrac></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>+</m:mo><m:mfrac><m:mn>1</m:mn><m:mn>2</m:mn></m:mfrac></m:mrow></m:mfenced></m:mrow><m:mo>+</m:mo><m:mrow><m:mfrac><m:mrow><m:mi>s</m:mi><m:mo>+</m:mo><m:mi>t</m:mi></m:mrow><m:mn>2</m:mn></m:mfrac><m:mo>⁢</m:mo><m:msup><m:mfenced open="(" close=")"><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>+</m:mo><m:mfrac><m:mn>1</m:mn><m:mn>2</m:mn></m:mfrac></m:mrow></m:mfenced><m:mn>2</m:mn></m:msup></m:mrow><m:mo>+</m:mo><m:mi mathvariant="normal">⋯</m:mi></m:mrow></m:mfenced></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>H</m:ci><m:apply><m:times/><m:ci>ℏ</m:ci><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>ω</m:ci><m:cn>0</m:cn></m:apply><m:apply><m:plus/><m:apply><m:divide/><m:apply><m:plus/><m:ci>s</m:ci><m:ci>t</m:ci></m:apply><m:cn>8</m:cn></m:apply><m:apply><m:times/><m:apply><m:minus/><m:cn>1</m:cn><m:apply><m:divide/><m:apply><m:plus/><m:ci>s</m:ci><m:ci>t</m:ci></m:apply><m:cn>2</m:cn></m:apply></m:apply><m:apply><m:plus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:apply><m:divide/><m:cn>1</m:cn><m:cn>2</m:cn></m:apply></m:apply></m:apply><m:apply><m:times/><m:apply><m:divide/><m:apply><m:plus/><m:ci>s</m:ci><m:ci>t</m:ci></m:apply><m:cn>2</m:cn></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:plus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:apply><m:divide/><m:cn>1</m:cn><m:cn>2</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply></m:apply><m:ci>⋯</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">58</span>)</td>
	</tr>
      
      </table>
    <p class="p">Apart from an irrelevant shift of the energy zero and a renormalization
of the frequency into <m:math display="inline"><m:semantics><m:mrow><m:mi>ω</m:mi><m:mo>=</m:mo><m:mrow><m:msub><m:mi>ω</m:mi><m:mn>0</m:mn></m:msub><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mn>1</m:mn><m:mo>-</m:mo><m:mfrac><m:mrow><m:mi>s</m:mi><m:mo>+</m:mo><m:mi>t</m:mi></m:mrow><m:mn>2</m:mn></m:mfrac></m:mrow></m:mfenced></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>ω</m:ci><m:apply><m:times/><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>ω</m:ci><m:cn>0</m:cn></m:apply><m:apply><m:minus/><m:cn>1</m:cn><m:apply><m:divide/><m:apply><m:plus/><m:ci>s</m:ci><m:ci>t</m:ci></m:apply><m:cn>2</m:cn></m:apply></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>
this Hamiltonian contains a quadratic anharmonicity
unless <m:math display="inline"><m:semantics><m:mrow><m:mi>s</m:mi><m:mo>=</m:mo><m:mrow><m:mo>-</m:mo><m:mi>t</m:mi></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>s</m:ci><m:apply><m:minus/><m:ci>t</m:ci></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>, i.e., unless <m:math display="inline"><m:semantics><m:mrow><m:mi>p</m:mi><m:mo>=</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>p</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:cn>1</m:cn></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>.
It is true that a quadratic anharmonicity will emerge even for the
P-type oscillator
(<m:math display="inline"><m:semantics><m:mrow><m:mi>p</m:mi><m:mo>=</m:mo><m:msup><m:mi>q</m:mi><m:mrow><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:msup></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>p</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>q</m:ci><m:apply><m:minus/><m:cn>1</m:cn></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>) as a residue of the fourth order term, but
it will be associated with a fourth order anharmonicity which may
well be inconsistent with the experimental spectrum of some
system of interest, such as a diatomic molecule.</p>
  </div>

  <div class="para" id="S7.p3">
    <p class="p">We finally point out that a coordinate and a conjugate
momentum can be defined for the deformed oscillator by means of
the relations
<m:math display="inline"><m:semantics><m:mrow><m:mover accent="true"><m:mi>x</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>=</m:mo><m:mrow><m:mfenced open="(" close=")"><m:mrow><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup><m:mo>+</m:mo><m:mi>a</m:mi></m:mrow></m:mfenced><m:mo>/</m:mo><m:msqrt><m:mn>2</m:mn></m:msqrt></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:ci>⌃</m:ci><m:ci>x</m:ci></m:apply><m:apply><m:divide/><m:apply><m:plus/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>a</m:ci></m:apply><m:apply><m:ci/><m:cn>2</m:cn></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math> and
<m:math display="inline"><m:semantics><m:mrow><m:mover accent="true"><m:mi>p</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>=</m:mo><m:mrow><m:mrow><m:mi>i</m:mi><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup><m:mo>-</m:mo><m:mi>a</m:mi></m:mrow></m:mfenced></m:mrow><m:mo>/</m:mo><m:msqrt><m:mn>2</m:mn></m:msqrt></m:mrow></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:ci>⌃</m:ci><m:ci>p</m:ci></m:apply><m:apply><m:divide/><m:apply><m:times/><m:ci>i</m:ci><m:apply><m:minus/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply><m:ci>a</m:ci></m:apply></m:apply><m:apply><m:ci/><m:cn>2</m:cn></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math>. Application of
Eq. (<a href="#S3.E20" title="Eq.20 in §3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">20</a>) with the choice <m:math display="inline"><m:semantics><m:mrow><m:mi>Q</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:ci>Q</m:ci><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math> results in
(for <m:math display="inline"><m:semantics><m:mrow><m:mrow><m:mi mathvariant="normal">ℏ</m:mi><m:mo>⁢</m:mo><m:msub><m:mi>ω</m:mi><m:mn>0</m:mn></m:msub></m:mrow><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:eq/><m:apply><m:times/><m:ci>ℏ</m:ci><m:apply><m:csymbol cd="ambiguous">subscript</m:csymbol><m:ci>ω</m:ci><m:cn>0</m:cn></m:apply></m:apply><m:cn>1</m:cn></m:apply></m:annotation-xml></m:semantics></m:math>)</p>
  <table class="equation" id="S7.E59">
      
      
	<tr valign="baseline" class="equation" id="S7.E59">
	<td class="eqpad"/>
	<td nowrap="yes" align="center" colspan="1"><m:math display="block"><m:semantics><m:mrow><m:mrow><m:mfenced open="[" close="]"><m:mrow><m:mover accent="true"><m:mi>x</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>,</m:mo><m:mover accent="true"><m:mi>p</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow></m:mfenced><m:mo>=</m:mo><m:mrow><m:mi>i</m:mi><m:mo>⁢</m:mo><m:mfenced open="[" close="]"><m:mrow><m:mi>a</m:mi><m:mo>,</m:mo><m:msup><m:mi>a</m:mi><m:mo>†</m:mo></m:msup></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mrow><m:mi>i</m:mi><m:mo>⁢</m:mo><m:mfenced open="[" close="]"><m:mrow><m:mn>1</m:mn><m:mo>+</m:mo><m:mrow><m:none/><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mi>s</m:mi><m:mo>+</m:mo><m:mi>t</m:mi><m:mo>-</m:mo><m:mrow><m:mfrac><m:msup><m:mfenced open="(" close=")"><m:mrow><m:mi>s</m:mi><m:mo>+</m:mo><m:mi>t</m:mi></m:mrow></m:mfenced><m:mn>2</m:mn></m:msup><m:mn>2</m:mn></m:mfrac><m:mo>⁢</m:mo><m:none/></m:mrow></m:mrow></m:mfenced><m:mo>⁢</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover></m:mrow><m:mo>+</m:mo><m:mrow><m:mfrac><m:mrow><m:msup><m:mi>s</m:mi><m:mn>2</m:mn></m:msup><m:mo>+</m:mo><m:mrow><m:mi>s</m:mi><m:mo>⁢</m:mo><m:mi>t</m:mi></m:mrow><m:mo>+</m:mo><m:msup><m:mi>t</m:mi><m:mn>2</m:mn></m:msup></m:mrow><m:mn>2</m:mn></m:mfrac><m:mo>⁢</m:mo><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>⁢</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mover accent="true"><m:mi>n</m:mi><m:mo>⌃</m:mo></m:mover><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:mfenced></m:mrow><m:mo>+</m:mo><m:mi mathvariant="normal">⋯</m:mi></m:mrow></m:mfenced></m:mrow></m:mrow><m:mo>,</m:mo></m:mrow><m:annotation-xml encoding="MathML-Content"><m:apply><m:ci/><m:apply><m:interval closure="closed"/><m:apply><m:ci>⌃</m:ci><m:ci>x</m:ci></m:apply><m:apply><m:ci>⌃</m:ci><m:ci>p</m:ci></m:apply></m:apply><m:eq/><m:apply><m:times/><m:ci>i</m:ci><m:apply><m:interval closure="closed"/><m:ci>a</m:ci><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>a</m:ci><m:ci>†</m:ci></m:apply></m:apply></m:apply><m:eq/><m:apply><m:times/><m:ci>i</m:ci><m:apply><m:plus/><m:cn>1</m:cn><m:apply><m:times/><m:ci/><m:apply><m:minus/><m:apply><m:plus/><m:ci>s</m:ci><m:ci>t</m:ci></m:apply><m:apply><m:times/><m:apply><m:divide/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:apply><m:plus/><m:ci>s</m:ci><m:ci>t</m:ci></m:apply><m:cn>2</m:cn></m:apply><m:cn>2</m:cn></m:apply><m:ci/></m:apply></m:apply><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply></m:apply><m:apply><m:times/><m:apply><m:divide/><m:apply><m:plus/><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>s</m:ci><m:cn>2</m:cn></m:apply><m:apply><m:times/><m:ci>s</m:ci><m:ci>t</m:ci></m:apply><m:apply><m:csymbol cd="ambiguous">superscript</m:csymbol><m:ci>t</m:ci><m:cn>2</m:cn></m:apply></m:apply><m:cn>2</m:cn></m:apply><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:apply><m:plus/><m:apply><m:ci>⌃</m:ci><m:ci>n</m:ci></m:apply><m:cn>1</m:cn></m:apply></m:apply><m:ci>⋯</m:ci></m:apply></m:apply></m:apply></m:annotation-xml></m:semantics></m:math></td>
	<td class="eqpad"/><td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">59</span>)</td>
	</tr>
      
      </table>
    <p class="p">from which follows the deformed uncertainty relation.</p>
  </div>

  <div class="para" id="S7.p4">
    <p class="p"><span style="" class="text bold">Acknowledgements</span></p>
  </div>

  <div class="para" id="S7.p5">
    <p class="p">One of the authors (JK) would like to thank the
Région Rhône-Alpes for a visiting fellowship
and the Institut
de Physique Nucléaire de Lyon for its kind hospitality.</p>
  </div>

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